Exercises: Hypothesis tests

A state senator claims that more than $60\%$ of her constituents support the new highway infrastructure bill. A local news station conducts a random phone poll of registered voters in her district to test this claim. Let $p$ represent the true proportion of constituents who support the bill. State the appropriate null and alternative hypotheses for this test.

A nationwide fast-food chain claims that $20\%$ of their drive-thru customers order a vegetarian meal. A skeptical franchise owner takes a random sample of $150$ drive-thru customers and finds that $36$ of them ordered a vegetarian meal. Calculate the test statistic ($z$) that should be used to test the chain's claim.

A quality control manager at a cereal factory conducts a hypothesis test to determine if the mean weight of their standard cereal boxes is less than the advertised $16$ ounces. The test results in a P-value of $0.034$. Using a significance level of $\alpha = 0.05$, state the decision (reject or fail to reject the null hypothesis) and provide the conclusion in the context of the problem.

A medical researcher is testing a newly developed drug designed to reduce blood pressure in patients with hypertension.

• $H_0$: The drug does not reduce blood pressure.

• $H_a$: The drug effectively reduces blood pressure.

Describe exactly what a Type I error would represent in the context of this medical study.

A physical therapist wants to know if there is a significant difference in flexibility between the left and right legs of professional soccer players. She measures the maximum extension angle of both the left and right legs for a random sample of $20$ players. To analyze this data, should the therapist use a two-sample t-test for a difference in means or a paired t-test for a mean difference? Briefly explain your choice.

A transportation researcher wants to compare the mean daily commute times of office workers in Chicago ($n_1 = 18$) and Seattle ($n_2 = 25$). Assuming the population distributions are roughly normal but the population variances are not assumed to be equal, what is the conservative number of degrees of freedom the researcher should use for a two-sample t-test?

An environmental protection agency is testing the water quality in a local lake to see if lead levels exceed the safe limit. The statistical power of their hypothesis test to detect a dangerously high level of lead is $0.88$. What is the probability of making a Type II error in this specific scenario?

A high school teacher wants to test if a new intensive SAT prep book improves students' math scores. She records the math scores of $15$ students before they use the book and their scores on a similar test after completing the book. Let $\mu_d$ be the true mean difference (After $-$ Before) in SAT math scores. State the appropriate null and alternative hypotheses to test if the book improves scores.

A commercial airline claims that less than $8\%$ of its domestic flights are delayed by more than $30$ minutes. A consumer rights group takes a random sample of $400$ domestic flights from this airline and finds that $26$ of them were delayed by more than $30$ minutes. Check if the conditions for inference are met, and if so, calculate the appropriate test statistic to evaluate the airline's claim.

A popular coffee shop chain claims their large iced coffee contains an average of $24$ fluid ounces. A skeptical customer believes they are being shortchanged and takes a random sample of $12$ large iced coffees. The sample yields a mean of $23.2$ fluid ounces with a standard deviation of $0.9$ fluid ounces. Assuming the volume of coffee is normally distributed, calculate the test statistic and identify the degrees of freedom for this hypothesis test.

A digital marketing firm wants to know if a new social media ad campaign has a higher click-through rate in New York than in Texas. They randomly sample users who were shown the ad in both states. In New York, $180$ out of $400$ sampled users clicked the ad. In Texas, $135$ out of $350$ sampled users clicked the ad. Calculate the pooled sample proportion ($\hat{p}_c$) and the test statistic ($z$) for testing the difference in proportions.

An automotive engineer is testing six cars to see if using premium gasoline improves fuel efficiency (in miles per gallon, mpg) compared to standard gasoline. Each car is tested once with standard gas and once with premium gas. The differences in fuel efficiency (Premium $-$ Standard) for the six cars are:

$1.2, \ 0.8, \ 1.5, \ 2.0, \ 0.5, \ 1.2$

Calculate the sample mean difference ($\bar{x}_d$), the sample standard deviation of the differences ($s_d$), and the appropriate test statistic.

A manufacturing plant produces heavy-duty steel cables used in elevator construction. The safe breaking strength of these cables is claimed to be at least $15,000$ lbs. A safety inspector tests a random sample of cables using the following hypotheses:

• $H_0: \mu = 15,000$ lbs

• $H_a: \mu < 15,000$ lbs

Describe the practical consequences of making a Type I error and a Type II error in this context. Which error poses a greater risk to public safety?

A sociologist uses statistical software to compare the mean annual salaries (in thousands of dollars) of male and female software engineers with $5$ years of experience. The software output for a two-sample t-test shows:

T-Value = 2.45, P-Value = 0.018, DF = 48.

Assuming the test is two-sided and the chosen significance level is $\alpha = 0.01$, what decision should the researcher make regarding the null hypothesis, and what is the conclusion in context?

The graph below shows the standard normal distribution for a two-sided hypothesis test regarding a population proportion. The calculated test statistic from the sample data is $z = 2.15$.

Explain what the shaded red regions represent in terms of the P-value. Based on standard normal probabilities, state whether the null hypothesis should be rejected at the $\alpha = 0.05$ significance level.

A city planner wants to test if the mean wait time at a specific downtown bus stop is greater than $15$ minutes. She randomly selects $12$ riders throughout the week and records their wait times. When she graphs the sample data, she finds that it is strongly skewed to the right with one clear outlier (a wait time of $45$ minutes). Explain why proceeding with a one-sample t-test for the population mean would not be statistically appropriate in this situation.

An environmental agency is testing whether the mean concentration of a specific pollutant in a river exceeds the legal safe limit of $50$ ppm. The hypotheses are $H_0: \mu = 50$ vs $H_a: \mu > 50$. The graph below illustrates the sampling distribution of the sample mean ($\bar{x}$) under the null hypothesis (blue curve) and under a specific alternative hypothesis where the true mean is $\mu = 55$ (green curve). The dashed orange line represents the critical value for rejecting $H_0$.

Identify which specific geometric areas under these curves correspond to the significance level ($\alpha$), the probability of a Type II error ($\beta$), and the power of the test. Then, explain how increasing the sample size $n$ would visually alter these curves and affect the power of the test, assuming $\alpha$ remains constant.

A sports scientist wants to determine if a newly engineered track shoe (Brand B) reduces the $100$-meter dash time compared to the current industry standard shoe (Brand A). He recruits $30$ collegiate sprinters for a study and considers two experimental designs:

• Design 1: Randomly assign $15$ athletes to wear Brand A and $15$ to wear Brand B. Each athlete runs one sprint.

• Design 2: Have all $30$ athletes run two sprints, one with Brand A and one with Brand B. The order of the shoes is randomized for each athlete, and sufficient rest is given between sprints.

Explain clearly why Design 2 is statistically more powerful and preferable to Design 1. Then, state the correct null and alternative hypotheses for the statistical test that would be used in Design 2.

A school district conducted a one-sample t-test to determine if the mean reading score of its 5th-grade students was significantly higher than the state average of $210$. The hypotheses were $H_0: \mu = 210$ vs $H_a: \mu > 210$. A random sample of $25$ students yielded a test statistic of $t = 2.064$. If the sample standard deviation was exactly $15$ points, what was the sample mean reading score ($\bar{x}$) of these $25$ students?

A tech company is running an A/B test on two versions of a checkout page to see which yields a higher conversion rate. Version A is shown to $n$ users and Version B is shown to a separate group of $n$ users (equal sample sizes). The pooled sample proportion of users who completed a purchase across both groups is $\hat{p}_c = 0.16$.

The test statistic for $H_0: p_A = p_B$ vs $H_a: p_A \neq p_B$ is calculated to be $z = -2.50$. If the observed difference in sample proportions is $\hat{p}_A - \hat{p}_B = -0.04$, calculate the sample size $n$ for each group. Show the algebraic steps used to find $n$.

A pharmaceutical company is concluding clinical trials for a new chemotherapy drug intended to treat a highly aggressive form of cancer.

• $H_0$: The new drug is not more effective than the current standard treatment.

• $H_a$: The new drug is more effective than the current standard treatment.

A Type I error would result in mass-producing and prescribing a drug that is no better than the standard, costing the healthcare system millions of dollars. A Type II error would result in discarding a highly effective drug, costing thousands of patients a chance at a longer life. Based on these severe real-world consequences, argue whether the researchers should choose a significance level of $\alpha = 0.01$ or $\alpha = 0.10$. Justify your reasoning by discussing the trade-off between the two types of errors.

A botanist is studying the effect of a new experimental fertilizer on tomato plant yield. She applies the standard fertilizer to one side of a garden plot and the new fertilizer to the other side. She measures the total yield (in pounds) of $11$ pairs of plants, carefully matched by soil quality and sunlight exposure. The dotplot below displays the differences in yield (New $-$ Standard) for the $11$ pairs.

The sample mean difference is $\bar{x}_d = 3.0$ lbs and the standard deviation is $s_d = 3.0$ lbs. Based on the dotplot, are the conditions for inference met for a small sample? If so, calculate the test statistic and determine if there is convincing evidence at the $\alpha = 0.05$ level that the new fertilizer increases yield on average.

A local school board funded an expensive pilot program intended to improve standardized test scores. After one year, a two-sample t-test comparing students in the program to those who were not yielded a P-value of $0.22$. During a public meeting, the board president stated, "This high P-value proves that the pilot program is completely ineffective and has zero impact on test scores."

Statistical software indicated that the power of this specific test to detect a meaningful improvement of $5$ points was only $0.35$. Explain why the board president's conclusion is statistically flawed, referencing both the correct interpretation of a non-significant P-value and the implications of the test's low power.

A consumer advocacy group wants to compare the continuous video playback battery life of two flagship smartphones, Model X and Model Y. They test a random sample of phones of each model under identical brightness and network conditions.

• Model X: $n = 35$, $\bar{x} = 14.2$ hours, $s = 1.6$ hours.

• Model Y: $n = 40$, $\bar{x} = 13.1$ hours, $s = 1.4$ hours.

State the hypotheses to test if there is a significant difference in mean battery life between the two models. Calculate the test statistic. Assuming a conservative degrees of freedom of $34$, the resulting two-tailed P-value is between $0.002$ and $0.005$. State your formal conclusion at the $\alpha = 0.01$ significance level in the context of the problem.