Inference for Means AP STA

Thirty randomly selected students took the calculus final. If the sample mean was 92 and the standard deviation was 9.4, construct a 99 percent confidence interval for the mean score of all students from which the sample was gathered.

• 89.08 < μ < 94.92

• 87.27 < μ < 96.73

• 87.29 < μ < 96.71

• 87.77 < μ < 96.23

A research group wishes to estimate the mean amount of time (in hours) that members of a fitness center spend exercising each week. They want to estimate the mean within a margin of error of 0.5 hours with a 95% level of confidence. Previous data suggests that the standard deviation of the population is 2.2. Which of the following is the smallest sample size they could use?

• 60

• 75

• 90

• 180

A 90% confidence interval for the average salary of all CEOs in the electronics industry was constructed using the results of a random survey of 45 CEOs. The interval was $139,048 to $154,144. Give the interpretation of the interval.

• 90% of the sampled CEOs have salaries that fell in the interval $139,048 to $154,144

• We are 90% confident that the mean salary of all CEOs in the electronics industry falls in the interval $139,048 to $154,144.

• There is a 90% chance that CEOs in the electronics industry have salaries that fall between $139,048 to $154,144

• We are 90% confident that the mean salary of the sampled CEOs falls in the interval $139,048 to $154,144.

You sample 12 bugs and find the sample mean is 2.40 cm. You are told the sample standard deviation is 0.2 cm. Find a 95% Confidence Interval.

Identify the Variables

• 12

• x-bar

• 2.201

• s

• 0.2

• t*

• 2.40

• n

Find the t* value for a 99% confidence interval with a sample of 23 people.

• 2.807

• 2.508

• 2.064

• 2.819

A significance test was performed to test 𝐻₀ : 𝜇 = 2 versus the alternative 𝐻_a: 𝜇 ≠ 2. A sample of size 28 produced a standardized test statistic of 𝑡 = 2.051. Assume all conditions for inference are met. Using technology the 𝑃-value is  (Round to 4 decimal places)   ___?

Answer explanation

tcdf(2.051, 1000, 27)

Double the answer to account for both tails of the distribution.

Using Table B and df = 27, 𝑡 = 2.051 falls between 0.025 and 0.05. Because this is a two-sided test, the 𝑃-value falls between 0.05 and 0.10.

A study of road rage asked separate random samples of 596 men and 523 women about their behavior while driving. Based on their answers, each respondent was assigned a road rage score on a scale of 0 to 20. Are the conditions for constructing a two-sample 𝑡 interval satisfied?

• Maybe; we have independent random samples, but we should look at the data to check Normality.

• Yes; the large sample sizes guarantee that the corresponding population distributions will be Normal.

• No; we don't know the population standard deviations.

• No; road rage scores on a scale from 0 to 20 can't be Normal.

• Yes; we have two independent random samples and large sample sizes.

Answer explanation

Because the values are on a scale of 1 to 20, the populations can’t be Normal. However, the large sample sizes justify the use of the two-sample 𝑡 test. Also, the data come from independent random samples that are both less than 10% of their respective populations.

A manufacturer claims that his tires average 40,000 miles. A test on 25 tires reveals that the mean life of a tire is 39,750 miles, with a standard deviation of 387 miles. You think that the tires average less than 40,000 miles. Test the Manufacturer’s claim at α = .01. Interpret your conclusion.

• There is enough evidence to suggest that the tires do not average 40,000 miles.

• There is not enough evidence to suggest that the tires do not average 40,000 miles.

An IQ test was given to a simple random sample of 75 students at a certain college. The sample mean score was 105.2. Scores on this test are known to have a standard deviation of 10. It is desired to construct a 90% confidence interval for the mean IQ scores of the students at the college. Find the critical value.

• 0.90

• 1.155

• 1.645

• ±1.645

A manufacturer claims that his tires average 40,000 miles. A test on 25 tires reveals that the mean life of a tire is 39,750 miles, with a standard deviation of 387 miles. You do not think that the tires average 40,000 miles. Test the Manufacturer’s claim at α = .01. Write the null and alternative hypothesis.