📝 Exam Questions — Probability & Statistics 2

A mobile phone repair technician claims that the proportion of successful screen repairs is 90%. A rival company suspects this claim is inaccurate and conducts a test. They send 20 phones with broken screens to the technician and observe that 16 of them are successfully repaired. (a) Formulate the null and alternative hypotheses for a two-tailed test. [2] (b) Determine the critical region for the number of successful repairs, X, at the 5% significance level. [4] (c) Evaluate whether the technician's claim is supported by the sample data. [3] (d) Explain what a Type I error would mean in the context of this problem. [2]

A manufacturer claims that 70% of their new light bulbs last for at least 1000 hours. A consumer organisation wants to test this claim. They select a random sample of 15 light bulbs and find that 4 of them last for at least 1000 hours. (a) Formulate the null and alternative hypotheses for this test. [2] (b) Calculate the probability of observing 4 or fewer successes if the null hypothesis is true, and determine if there is sufficient evidence to reject the null hypothesis at the 5% significance level. [5]

A statistician is conducting a hypothesis test on a population parameter, assuming the test statistic follows a standard normal distribution. Fig 1.1 illustrates the standard normal distribution curve. Fig 1.1 (a) Use Fig 1.1 to identify the critical value for a one-tailed test at the 2.5% significance level, where the rejection region is in the upper tail. [4] (b) Explain how the critical region would be represented on the graph if it was a two-tailed test at the same significance level. [5]

A manufacturer claims that less than 5% of their products are defective. A quality control test is performed to verify this claim at a 10% significance level. (a) Identify the probability of a Type I error for this test. [4]

Fig 1.1 shows a normal distribution curve with a shaded critical region in the right tail, starting at a critical value of 1.645. The x-axis represents the test statistic value. (a) Describe the relationship between the test statistic and the critical region in a hypothesis test, referring to Fig 1.1. [4] (b) Outline how the value of the test statistic helps in making a decision about the null hypothesis. [4]

A manufacturer claims that the average lifespan of their light bulbs is 1000 hours. A consumer group suspects this claim is incorrect and conducts a hypothesis test to check if the average lifespan is different from 1000 hours. (a) Sketch a diagram to illustrate the critical region for a two-tailed hypothesis test at the 5% significance level for a normal distribution. [4] (b) Explain how the critical value is used to determine if a test statistic falls within the critical region. [4]

Hypothesis testing is a fundamental tool in scientific research, guiding decisions in fields from medicine to engineering. The initial formulation of hypotheses is crucial. (a) Discuss the potential implications of incorrectly formulating the null and alternative hypotheses in a medical drug trial. [6] (b) Identify two scenarios where a two-tailed hypothesis test would be appropriate, giving reasons. [4]

A factory produces electronic components, and historically, 5% of these components are defective. A new manufacturing process is introduced, and the quality control manager wants to test if the proportion of defective items has increased. A random sample of 200 components produced by the new process is taken. Fig 1.1 illustrates a normal distribution curve that could be used to model the number of defective items. (a) Formulate the null and alternative hypotheses for this test. [2] (b) Show that the conditions for using a normal approximation to the binomial distribution are met and calculate the mean and variance of the approximating normal distribution. [4] (c) Find the critical region for the number of defective items, X, at the 2.5% significance level for a one-tailed test. [4] (d) Discuss the implications of the finding if 18 defective items were found in the sample. [2]

In a study of human reaction times, a researcher is testing whether a new training program significantly improves performance. The test statistic is assumed to follow a standard normal distribution. (a) Fig 1.2 shows a standard normal distribution curve with a shaded region in the lower tail. Analyse Fig 1.2 to determine the critical value for a one-tailed test with a rejection region of 10% in the lower tail. [5] (b) Sketch a normal distribution curve and clearly label the critical values and rejection regions for a two-tailed test at the 1% significance level. [7]

A pharmaceutical company is developing a new drug to treat high blood pressure. Before conducting extensive trials, they need to establish the initial hypotheses for their study. (a) Formulate the null hypothesis (H0) for a claim that a new drug has no effect on blood pressure. [3] (b) Explain why it is important for H0 to be a precise statement that can be tested. [4]

A researcher is planning a hypothesis test to investigate a claim about a population proportion, p. (a) Identify the key characteristic that distinguishes a two-tailed test from a one-tailed test. [2] (b) State the alternative hypothesis, H1, for a two-tailed test if the null hypothesis is H0: p = 0.3. [3]

A coin is tossed 10 times. A student claims the coin is biased towards tails. In these 10 tosses, 3 heads are observed. (a) Formulate the null and alternative hypotheses for testing the student's claim. [2] (b) Calculate the probability of obtaining 3 or fewer heads in 10 tosses if the coin is fair. [3]

When conducting a hypothesis test, there is always a risk of making an incorrect decision regarding the null hypothesis. (a) Define what is meant by a Type I error in the context of hypothesis testing. [3] (b) Give an example of a real-world scenario where a Type I error could occur. [3]

In the process of developing new pharmaceuticals, extensive hypothesis testing is conducted to evaluate the efficacy and safety of potential drug candidates. Researchers must carefully consider the implications of errors in their statistical conclusions. (a) Analyse the trade-off between Type I and Type II errors when designing a hypothesis test. How does adjusting the significance level affect both probabilities? [6] (b) Compare the consequences of making a Type I error versus a Type II error in the context of testing a new medical treatment for a serious illness. [5]

In a statistical study, a researcher is interested in whether a certain population parameter has changed from a known value, without specifying a direction of change. (a) Calculate the critical values for a two-tailed hypothesis test at the 10% significance level, assuming a standard normal distribution. [4] (b) Determine the acceptance region for this test. [4]

A company claims that 70% of its customers are satisfied with their service. A new customer satisfaction program is implemented, and the company wants to test if the proportion of satisfied customers has increased. A random sample of 120 customers is surveyed, and 92 of them report being satisfied. Fig 1.2 shows a standard normal distribution curve with a critical value for a one-tailed test. (a) State the null and alternative hypotheses for this test. [2] (b) Calculate the Z-score for the observed number of successes, using continuity correction. [4] (c) Determine if there is sufficient evidence at the 5% significance level to suggest the new process is more effective, referencing the sketch in Fig 1.2. [3]

A medical researcher claims that a new drug is effective in reducing symptoms for more than 75% of patients. In a clinical trial, the drug was administered to 120 patients, and 98 of them experienced a reduction in symptoms. (a) State the null and alternative hypotheses. [2] (b) Explain why a normal approximation can be used here. [2] (c) Calculate the test statistic using the normal approximation, including continuity correction. [4] (d) Determine if there is evidence at the 1% significance level to support the researcher's claim. [2]

A pharmaceutical company is testing a new drug designed to lower blood pressure. The current drug leads to a mean blood pressure reduction of 50 units. The new drug is hypothesised to provide a greater reduction. Fig 1.3 illustrates the distributions under the null and alternative hypotheses, along with a critical value. Fig 1.3 displays two normal distribution curves. The first curve, representing H0, is centered at a mean of 50 with a standard deviation of 4. The second curve, representing H1, is centered at a mean of 55 with a standard deviation of 4. A vertical dashed line is drawn at x = 53, representing the critical value. The area under the H0 curve to the right of 53 is shaded (alpha). The area under the H1 curve to the left of 53 is also shaded (beta). (a) Identify the region in Fig 1.3 that represents the probability of a Type I error. [3] (b) Calculate the probability of a Type II error using the information from Fig 1.3, given that the true population mean is 55. The critical value for rejection is 53. [6]

A quality control manager is testing a new manufacturing process to see if it reduces the proportion of defective items. They decide to use a hypothesis test based on a standard normal distribution for their test statistic. (a) Determine the critical value(s) for a two-tailed hypothesis test at the 5% significance level, for a test statistic that follows a standard normal distribution. [4] (b) Explain how the critical value(s) change if the significance level is reduced to 1% for the same test. [4]

A quality control manager at a manufacturing plant is monitoring the average weight of a certain product. Historically, the mean weight has been 50 grams. A sample of 100 products is taken to check if the mean weight has changed. (a) Evaluate the critical values and rejection region for a two-tailed test of the population mean, μ, at the 1% significance level, given a sample of 100 observations from a normal distribution with known standard deviation σ = 5, and H0: μ = 50. Show all steps. [8] (b) Interpret what it means if a calculated sample mean of 51.5 falls within the rejection region. [4]

A school introduces a new teaching method with the claim that it will improve student test scores. To assess this claim, a hypothesis test is planned. (a) Formulate the null and alternative hypotheses for a claim that a new teaching method improves student test scores. [4] (b) Describe the type of error that would occur if the new teaching method actually has no effect, but the hypothesis test concludes it does. [4]

A quality control manager is monitoring the weight of cereal boxes, which are designed to have a mean weight of 50 grams with a standard deviation of 5 grams. A one-tailed hypothesis test is to be conducted to see if the weight has increased. (a) Calculate the critical value for a one-tailed hypothesis test (upper tail) at the 1% significance level, given that the test statistic follows a normal distribution with mean 50 and standard deviation 5. [4] (b) Interpret what it means if a calculated test statistic of 62 is obtained in this test. [3]

A nutritionist is testing a new diet plan. She sets up a hypothesis test to determine if the diet leads to a significant change in weight. (a) Define what is meant by the critical region (or rejection region) in a hypothesis test. [2] (b) Identify the relationship between the critical region and the significance level. [3]

A medical researcher is conducting a hypothesis test to determine if a new drug has a different success rate than a placebo. They collect data from a sample of patients and will use this data to calculate a test statistic. (a) Identify the primary purpose of a test statistic in a hypothesis test. [2] (b) Explain why the observed number of successes in a sample is often used as the test statistic for a binomial hypothesis test. [4]

Studies suggest that 10% of the world's population is left-handed. A school administrator suspects that the proportion of left-handed students in their school is different from the national average. They take a random sample of 150 students from the school and find that 21 of them are left-handed. (a) Calculate the expected number of left-handed students in the sample if the national average holds true. [3] (b) Compare the observed number of left-handed students with the expected number. [2] (c) Explain how this comparison informs the choice of test statistic. [2]

A team of agricultural scientists is testing new farming techniques and products. They are particularly interested in understanding how different interventions affect crop yields. (a) Explain the difference between a one-tailed and a two-tailed hypothesis test. [3] (b) Formulate the null and alternative hypotheses for a study investigating if a new fertilizer increases crop yield. [4]

In the field of medical research, new treatments often undergo rigorous testing to determine their effectiveness. A key part of this process involves making formal statements about the treatment's effect. (a) Define what is meant by the null hypothesis, H0. [2] (b) Define what is meant by the alternative hypothesis, H1. [2] (c) State the purpose of conducting a hypothesis test. [1]

In quality control processes, manufacturers often use hypothesis tests to ensure their products meet certain standards. A crucial element in these tests is deciding how strong the evidence needs to be to reject a claim. (a) Define what is meant by the significance level in a hypothesis test. [2] (b) State why a low significance level might be preferred in some hypothesis tests. [2]

A manufacturer claims that the mean weight of their product is 10 kg. A quality control manager conducts a hypothesis test to check this claim. The distribution of product weights is assumed to be normal with a standard deviation of 2 kg. The results of the test are visualised in Fig 1.2. Fig 1.2 shows a normal distribution curve with the x-axis ranging from 4 to 16. The mean is at 10. Two vertical dashed lines are drawn at x = 6.08 and x = 13.92, indicating the boundaries of the central 95% region. The areas in the tails beyond these lines are shaded. (a) Determine the critical values from Fig 1.2 for a two-tailed test at the 5% significance level, given a sample mean of 10 and a standard deviation of 2. [4] (b) Discuss the decision if a calculated test statistic is found to be 2.1, with reference to the critical values. [3]

A medical researcher is testing a new drug and wants to understand the potential errors in their hypothesis testing. (a) Explain the relationship between the significance level of a test and the probability of a Type I error. [4] (b) Calculate the probability of a Type II error for a test where the null hypothesis H0: p = 0.5 is tested against H1: p > 0.5 at the 5% significance level, with a sample size of 20, given that the true proportion is p = 0.7. Use a binomial distribution. [5]

In a clinical trial, a new drug is being tested to see if it reduces blood pressure. The null hypothesis states that the drug has no effect on blood pressure. (a) Define a Type I error in the context of hypothesis testing. [2] (b) State the relationship between the significance level and the probability of a Type I error. [3]

A coin is tossed many times, and the proportion of heads is recorded. The distribution of this proportion is shown in Fig 1.3. (a) Fig 1.3 shows a normal distribution curve. Identify the null hypothesis being tested in the context of the distribution shown in Fig 1.3. [3] (b) Relate the central peak of the distribution to the value proposed by the null hypothesis. [3]

A company claims that 70% of its new energy-efficient light bulbs last for more than 5000 hours. A consumer organisation decides to test this claim by taking a random sample of 15 light bulbs. They observe that 9 of these light bulbs last for more than 5000 hours. Fig 1.1 shows the probability distribution for the number of light bulbs, X, lasting more than 5000 hours in a sample of 15, assuming the company's claim is true. (a) State the null and alternative hypotheses for this one-tailed test. [2] (b) Find the critical region for the number of successes, X, at the 10% significance level. [5] (c) Conclude whether the company's claim is supported by the data. [2]

A pharmaceutical company is conducting a medical trial for a new vaccine designed to prevent a common disease. The current vaccine has a known efficacy of 80%. The company wants to test if the new vaccine has a different efficacy. (a) Discuss the implications of incorrectly formulating the null hypothesis in this medical trial. [5] (b) Evaluate why the null hypothesis is always assumed to be true at the start of a hypothesis test. [5]

The choice between a one-tailed and a two-tailed hypothesis test is crucial and impacts the critical region. (a) Describe how the critical region for a one-tailed test differs from that of a two-tailed test at the same significance level. [3] (b) Sketch two normal distribution curves, one illustrating a one-tailed critical region (right-tailed) and the other a two-tailed critical region, both for a 5% significance level. Label the critical regions and the mean. [5]

A pharmaceutical company is developing new medications and needs to rigorously test their efficacy, which involves careful consideration of hypothesis testing approaches. (a) Discuss the implications of incorrectly choosing between a one-tailed and a two-tailed test on the conclusion of a hypothesis test. [6] (b) Justify the choice of a one-tailed test in a scenario where a new drug is being tested to see if it *reduces* blood pressure, compared to an existing drug. [4]

In the context of a statistical investigation, a researcher is often interested in determining if there is evidence to support a new claim or a change from an established belief. (a) State the purpose of the null hypothesis in a statistical test. [2] (b) Explain why the null hypothesis is often a statement of 'no effect' or 'no difference'. [3]

An airline claims that 50% of its passengers choose to pay for extra legroom. To test this claim, a random sample of 50 passengers is taken. It is found that 20 of these passengers paid for extra legroom. Fig 1.2 shows a normal distribution curve with a mean of 25. (a) Justify the use of a normal approximation to the binomial distribution in this scenario. [2] (b) Calculate the p-value for the observed number of successes and test the hypothesis at the 5% significance level, stating your conclusion. [6]

In hypothesis testing, decisions are made about a population parameter based on sample data. (a) Define the term 'one-tailed hypothesis test'. [2] (b) Explain when a one-tailed test would be appropriate to use, providing an example. [3]

A medical researcher is investigating whether a new treatment has an impact on patient recovery time. After collecting data, they need to formulate hypotheses to test their findings. (a) Define what is meant by the alternative hypothesis, H1. [2] (b) Give two examples of situations where an alternative hypothesis would be one-tailed. [3]

In the context of hypothesis testing, various terms are used to describe the process of making a decision about a population parameter. One such term is the 'test statistic'. (a) Define the term 'test statistic' in the context of hypothesis testing. [2] (b) State three key pieces of information needed to calculate a test statistic for a binomial distribution. [3]

A researcher is investigating whether the average height of adult males in a particular country has changed from the historical average of 175 cm. They collect a sample of adult males and measure their heights. (a) Identify whether the researcher's claim that 'the average height of adult males has increased' requires a one-tailed or two-tailed alternative hypothesis. [3] (b) Formulate the null and alternative hypotheses for this claim, assuming the historical average height was 175 cm. [5]

A quality control manager at a manufacturing plant is testing whether the average weight of a product has changed from the specified 500 grams. (a) Compare the choice between a one-tailed and a two-tailed alternative hypothesis, including scenarios where each would be appropriate. [6] (b) Justify why it is generally considered more conservative to use a two-tailed test if there is no strong prior reason to expect a specific direction of effect. [5]

A company claims that the mean weight of their cereal boxes is 500g. A consumer group suspects it is less. They decide to test this claim. They set up the hypotheses H0: μ = 500 against H1: μ < 500 and conduct a test at a 5% significance level using a sample of 40 boxes. The population standard deviation is known to be 20g. (a) If the true mean weight is 490g, calculate the probability of a Type II error. [6] (b) Discuss one practical consequence of making a Type II error in this scenario. [3]

A researcher is investigating whether the proportion of defective items produced by a new machine is greater than the standard 0.5. A hypothesis test is performed with H0: p = 0.5 and H1: p > 0.5 at a 5% significance level. (a) If a sample of 20 items is taken and the critical region is defined as X ≥ 14 (where X is the number of defective items), calculate the probability of a Type II error if the true proportion of defective items is actually p = 0.6. [5] (b) Interpret the meaning of your calculated Type II error probability in the context of this test. [3]

A company claims that its new energy drink improves reaction times. A researcher decides to test this claim using a hypothesis test. (a) Calculate the probability of a Type I error for a test conducted at the 1% significance level. [3] (b) Explain why increasing the significance level affects the probability of a Type I error. [4]

A manufacturer claims that the mean lifespan of a certain electronic component is 1000 hours. A consumer group suspects this claim is incorrect, either higher or lower. (a) Sketch a normal distribution curve showing the critical region for a two-tailed test at the 5% significance level. Label the axes and the critical values. [4] (b) Shade the region on your sketch that represents the probability of a Type I error. [4]

A pharmaceutical company is testing a new drug designed to increase the proportion of patients who react positively to a treatment. The current proportion is known to be 0.1. The hypothesis test is set up as H0: p = 0.1 (no effect) and H1: p > 0.1 (drug has effect). (a) A sample of 100 patients is used, and the critical region for the number of positive reactions (X) is X ≥ 15. Determine the probability of a Type II error if the true proportion of patients who react positively is 0.12. Use the normal approximation to the binomial distribution with continuity correction. [6] (b) Discuss how increasing the sample size would affect the probability of a Type II error, assuming the significance level remains constant. Illustrate your answer with a reference to the normal distribution curve. [6]

A pharmaceutical company is developing a new drug to treat a life-threatening disease. They need to decide on the significance level for testing the drug's effectiveness. (a) Discuss the implications of setting a very low significance level on the probability of a Type I error and the risk of failing to detect a true effect. [5] (b) Evaluate a scenario where a medical test for a rare disease has a 0.1% Type I error probability. What does this mean for a healthy individual? [5]

In the context of hypothesis testing, understanding different types of errors is crucial for interpreting results. (a) Define a Type II error. [2] (b) Explain why the probability of a Type II error (beta) is often harder to calculate than the probability of a Type I error. [3]

The power of a hypothesis test is a crucial concept in determining the effectiveness of a test. Fig 1.2 illustrates a power curve for a hypothesis test, showing the probability of rejecting the null hypothesis (power) against different true values of the population parameter. (a) Analyse how the probability of a Type II error changes as the true population parameter deviates further from the null hypothesis value, using Fig 1.2. [5] (b) Using Fig 1.2, suggest two ways to decrease the probability of a Type II error, explaining the effect of each on the power curve. [6]

A statistician performs a one-tailed hypothesis test. The results of the test are visualised in Fig 1.1. Fig 1.1 (a) From Fig 1.1, identify the significance level used for the one-tailed hypothesis test shown. [3] (b) Explain what the shaded region in Fig 1.1 represents in terms of a Type I error. [4]

Fig 1.4 shows a bar chart of the probability distribution for X ~ B(15, 0.4). The shaded bars indicate the critical region for a left-tailed hypothesis test. (a) Identify the critical value from Fig 1.4 for the left-tailed test. (b) Explain how the critical value in Fig 1.4 is determined, considering a significance level of 5% and the probabilities shown. (c) Calculate the actual significance level represented by the critical region in Fig 1.4.

Fig 1.28 shows a bar chart representing the probability distribution of the number of successes for X ~ B(30, 0.5). (a) Identify the critical region for rejecting the null hypothesis from Fig 1.28. [2] (b) If the true probability of success is 0.4 (instead of the null hypothesis's p=0.5), calculate the probability of a Type II error for the test shown in Fig 1.28, using the given acceptance region and approximating with a normal distribution for the new true parameter. [5] (c) Analyze how decreasing the significance level (making the critical regions smaller) would impact the probability of a Type II error, referring to Fig 1.28. [4]

Fig 1.23 shows a normal distribution curve for X ~ N(72, 28.8) with a shaded rejection region on the right tail. a) Determine the critical value for a right-tailed test at the 5% significance level from the normal distribution shown in Fig 1.23, given the mean and variance. b) If a sample yielded a value of X=85, calculate the corresponding z-score using the mean and variance provided by Fig 1.23, applying continuity correction. c) Evaluate whether the null hypothesis would be rejected for X=85 at the 5% significance level, based on the critical value from part (a).

Fig 1.19 shows a normal distribution curve for X ~ N(72, 28.8). (a) Formulate the null hypothesis (H0) for a test that would yield the distribution and rejection region shown in Fig 1.19. (b) Interpret the meaning of the unshaded region in Fig 1.19 in the context of accepting or rejecting H0. (c) Suggest a scenario where a right-tailed test, as depicted in Fig 1.19, would be appropriate.

Fig 1.32 shows a bar chart representing the probability distribution of the number of sixes rolled in 16 rolls of a fair die, X ~ B(16, 1/6). (a) Formulate the null hypothesis (H0) for a test that would result in the distribution and critical region shown in Fig 1.32. (b) Calculate the actual significance level of the test shown in Fig 1.32, using the probabilities for the shaded bars. (c) Conclude whether a sample result of 5 sixes would lead to the rejection of H0 at the significance level shown in Fig 1.32.

Fig 1.5 shows the probability distribution of the number of sixes rolled in 16 rolls of a fair die. (a) Formulate the null hypothesis (H0) based on the assumption of a fair die, as suggested by the distribution in Fig 1.5. (b) Interpret what the unshaded region in Fig 1.5 represents in the context of a hypothesis test.

Fig 1.26 shows a table illustrating the outcomes of a hypothesis test. (a) Identify the cell in Fig 1.26 that represents a Type II error. [2] (b) Define a Type I error based on the information presented in Fig 1.26. [3]

Fig 1.12 shows a table illustrating the possible outcomes of a hypothesis test. (a) Identify the cell in Fig 1.12 that represents a Type I error. [2] (b) Explain the conditions under which a Type II error occurs, referring to Fig 1.12. [3] (c) Describe the consequences of a Type I error in a medical context where the null hypothesis states a drug has no effect. [3]

Fig 1.15 shows a bar chart representing the probability distribution of the number of sixes rolled in 16 rolls of a fair die. (a) State the null hypothesis suggested by the distribution for a fair die in Fig 1.15. [2] (b) Determine the critical value for the number of sixes for a right-tailed test, given the shaded region in Fig 1.15. [3] (c) Evaluate the probability of observing 6 or more sixes, as indicated by the shaded region in Fig 1.15. [3]

Fig 1.25 shows a bar chart representing the probability distribution of the number of successes for X ~ B(20, 0.5). (a) Determine the critical values for the two-tailed test depicted in Fig 1.25. [2] (b) Calculate the total probability of observing a value within the critical region, based on the shaded areas in Fig 1.25. [3] (c) Interpret what the unshaded region in Fig 1.25 represents in the context of a two-tailed hypothesis test. [3]

Fig 1.24 shows a normal distribution curve for X ~ N(104, 49.92) with a shaded rejection region on the left tail. a) Identify the type of test (one-tailed or two-tailed) shown in Fig 1.24. b) Explain how the location of the shaded region in Fig 1.24 indicates the direction of the alternative hypothesis.

Fig 1.10 shows a bar chart representing the probability distribution for the number of sixes rolled in 16 rolls of a fair die. (a) Identify whether Fig 1.10 illustrates a one-tailed or a two-tailed test. [2] (b) Justify your answer to part (a) by referring to the shaded regions in Fig 1.10. [3]

Fig 1.3 shows the probability distribution for the number of times a spinner lands on red in ten spins, X ~ B(10, 0.25). The shaded bars represent the rejection region for a hypothesis test. (a) Determine the critical value for the number of successes based on the shaded rejection region in Fig 1.3. (b) State the null hypothesis for a test where the critical region is shown in Fig 1.3, assuming it's a test for an increased proportion. (c) Calculate the exact probability of a Type I error for the critical region shown in Fig 1.3, given the probabilities for each bar.

Fig 1.6 shows the probability distribution for the number of successes, X, in a binomial distribution with n=20 and p=0.6, with a shaded rejection region. (a) State the alternative hypothesis (H1) that corresponds to the shaded rejection region in Fig 1.6, assuming a test for a decreased proportion. (b) Explain why the rejection region in Fig 1.6 is only on the left tail, based on the nature of the alternative hypothesis.

Fig 1.18 shows a normal distribution curve for X ~ N(104, 49.92). (a) Determine the critical value for a left-tailed test at the 5% significance level from the normal distribution shown in Fig 1.18, given the mean and variance. (b) If the true mean of the distribution was 90 instead of 104, calculate the probability of accepting H0 (i.e., not rejecting H0) based on the critical value identified in part (a). (c) Discuss the implications of shifting the critical value to the left (decreasing it) on the probabilities of Type I and Type II errors.

Fig 1.17 shows a bar chart representing the probability distribution of the number of sixes rolled in 16 rolls of a fair die, X ~ B(16, 1/6). (a) Describe the critical region shown in Fig 1.17 in terms of the number of sixes. (b) State the probability associated with the critical region shown in Fig 1.17. (c) Relate the shaded region in Fig 1.17 to the concept of a Type I error.

Fig 1.8 shows the probability distribution for the number of times a spinner lands on red in ten spins, X, where X ~ B(10, 0.5), with two shaded critical regions. (a) Calculate the exact significance level of the test shown in Fig 1.8 by summing the probabilities of the shaded regions. (b) Determine the critical values for the number of reds based on the shaded regions in Fig 1.8. (c) State the type of hypothesis test (one-tailed or two-tailed) represented by Fig 1.8.

Fig 1.7 shows a normal distribution curve representing a hypothesis test, with a shaded rejection region. (a) Identify the critical value for the normal approximation shown in Fig 1.7. (b) If a test statistic of -1.8 is calculated (in standard normal units), compare this value to the critical value for the standardised variable (Z) corresponding to the critical value identified in part (a). (c) Conclude whether the null hypothesis would be rejected or accepted at the significance level shown in Fig 1.7, given a test statistic of -1.8.

Fig 1.13 shows two bar charts representing different critical regions for a hypothesis test based on a binomial distribution X ~ B(30, 0.5). (a) Calculate the probability of a Type I error for Graph A in Fig 1.13 by summing the probabilities of the shaded regions. [3] (b) Compare the probability of a Type I error for Graph A with that for Graph B in Fig 1.13. [3] (c) Discuss how changing the acceptance region from Graph A to Graph B in Fig 1.13 would affect the probability of a Type II error. [4]

Fig 1.29 shows a table illustrating the possible outcomes of a hypothesis test. (a) State the probability of a Type I error for the test shown in Fig 1.29. (b) Identify the type of error that occurs when a false null hypothesis is accepted, referring to Fig 1.29.

Fig 1.16 shows a bar chart representing the probability distribution of the number of times a spinner lands on red in ten spins, assuming the spinner is fair (P(red) = 0.5). (a) Identify the critical region for the two-tailed test shown in Fig 1.16. [2] (b) Calculate the actual significance level for the test illustrated in Fig 1.16 by summing the probabilities of the shaded regions. [3] (c) Discuss how the choice of critical values in Fig 1.16 ensures a balanced two-tailed test if the significance level is 5%. [4]

Fig 1.31 shows a normal distribution curve with a shaded rejection region for a right-tailed test. (a) Identify the critical value for the right-tailed test shown in Fig 1.31. (b) Interpret what the critical value in Fig 1.31 signifies in the context of a hypothesis test, assuming a 5% significance level.

Fig 1.20 shows a bar chart representing the probability distribution of the number of sixes rolled in 16 rolls of a fair die, X ~ B(16, 1/6). (a) State the alternative hypothesis (H1) for the hypothesis test depicted in Fig 1.20. (b) Explain why the rejection region is located on the right tail in Fig 1.20.

Fig 1.22 shows a bar chart representing the probability distribution of the number of times a spinner lands on red in ten spins, X ~ B(10, 0.5). Two critical regions are shaded: X <= 1 and X >= 9. a) Determine the critical values for the two-tailed test based on the shaded regions in Fig 1.22. b) Calculate the total probability of a Type I error for the test shown in Fig 1.22. c) Evaluate whether a significance level of 5% is achieved by the critical regions shown in Fig 1.22. d) Compare the acceptance region for this test with one that would result from a one-tailed test at the 5% level, testing H0: p=0.5 against H1: p>0.5.

Fig 1.21 shows a bar chart representing the probability distribution of the number of times a spinner lands on red in ten spins, X ~ B(10, 0.25). a) Determine the critical value for the number of successes from Fig 1.21. b) Interpret what it means if a calculated test statistic falls within the shaded region in Fig 1.21. c) Calculate the probability of observing exactly 6 successes, as shown by the bar in the critical region in Fig 1.21.

Fig 1.11 shows a bar chart representing the probability distribution of the number of successes for a binomial distribution X ~ B(20, 0.5). (a) Determine the critical values for the number of successes for the two-tailed test shown in Fig 1.11. [2] (b) Calculate the total significance level for the test shown in Fig 1.11 by summing the probabilities of both shaded regions. [4]

Fig 1.14 shows a bar chart representing the probability distribution for a hypothesis test where the null hypothesis states that the probability of success, p, is 0.5 for a binomial distribution X ~ B(20, p). (a) Identify the acceptance region for the test shown in Fig 1.14. [2] (b) If the true probability of success is 0.7 instead of 0.5, calculate the probability of a Type II error for the test shown in Fig 1.14, using the given acceptance region and approximating with a normal distribution for the new true parameter. [5] (c) Explain the trade-off between Type I and Type II errors, referencing the critical region shown in Fig 1.14. [4]

Fig 1.9 shows a normal distribution curve that approximates a binomial distribution with parameters n=120 and p=0.6. (a) Estimate the mean and variance of the normal distribution shown in Fig 1.9 from its curve. [3] (b) Given that the shaded region in Fig 1.9 represents a 5% significance level, calculate the critical value for a right-tailed test, applying continuity correction. Assume the mean and variance from part (a) are exact for calculation purposes. [4] (c) Discuss the validity of using a normal approximation for a binomial distribution with the parameters implied by Fig 1.9. [3]

Fig 1.2 shows two bar charts, Graph A and Graph B, representing the probability distribution for X ~ B(10, 0.2). (a) Estimate the significance level used for the right-tailed test shown in Graph A by reading the cumulative probability of the shaded region. (b) Explain the implication of using a smaller significance level on the critical region, referring to Fig 1.2.

Fig 1.1 shows the probability distribution for the number of sixes obtained in 16 rolls of a fair die, where X ~ B(16, 1/6). (a) Identify the critical region for rejecting the null hypothesis based on the shaded area in Fig 1.1. (b) Define what the significance level represents in the context of the hypothesis test shown in Fig 1.1.