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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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 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 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 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]

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 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]

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).

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]

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]

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]

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.

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]

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 1.18 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.