Ch. 8 - Hypothesis Testing with Two Samples

Larson - Elementary Statistics: Picturing the World 8th Edition

Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook

Larson 8th Edition

Ch. 8 - Hypothesis Testing with Two Samples

Problem 8.CR.14a

Chapter 8, Problem 8.CR.14a

[APPLET] The annual earnings (in dollars) for 30 randomly selected locksmiths are shown below. Assume the population is normally distributed. (Adapted from Salary.com)

48,69446,85642,91261,67271,11254,861

69,45471,84159,75169,61254,28452,166

66,36048,16465,27235,25061,12765,397

58,92558,91659,01753,07045,19969,941

69,49257,08553,82952,69268,29853,792

Construct a 95% confidence interval for the population mean annual earnings for locksmiths.

Verified step by step guidance

Step 1: Identify the key components of the problem. We are tasked with constructing a 95% confidence interval for the population mean annual earnings of locksmiths. The sample size (n) is 30, and the population is assumed to be normally distributed. The sample data is provided, so we will calculate the sample mean (x̄) and sample standard deviation (s).

Step 2: Calculate the sample mean (x̄). To do this, sum all the earnings data provided and divide by the sample size (n = 30). Use the formula:

x̄ = \frac{\sum x}{n}

, where Σx is the sum of all data points.

Step 3: Calculate the sample standard deviation (s). Use the formula:

s = \sqrt{\frac{\sum (x - x̄)^{2}}{n - 1}}

, where x̄ is the sample mean, x represents each data point, and n is the sample size.

Step 4: Determine the critical value (t*) for a 95% confidence level. Since the sample size is 30, use a t-distribution table with degrees of freedom (df = n - 1 = 29) to find the t* value corresponding to a 95% confidence level.

Step 5: Construct the confidence interval using the formula:

[x̄ - t * \frac{s}{\sqrt{n}}, x̄ + t * \frac{s}{\sqrt{n}}]

, where x̄ is the sample mean, t* is the critical value, s is the sample standard deviation, and n is the sample size.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the population parameter with a specified level of confidence, typically expressed as a percentage. For example, a 95% confidence interval suggests that if we were to take many samples and construct intervals in the same way, approximately 95% of those intervals would contain the true population mean.

Normal Distribution

Normal distribution is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. In the context of this question, assuming the population of locksmiths' earnings is normally distributed allows us to use specific statistical methods to calculate the confidence interval for the mean.

Sample Mean and Standard Deviation

The sample mean is the average of a set of values, calculated by summing all observations and dividing by the number of observations. The sample standard deviation measures the amount of variation or dispersion in a set of values. Both the sample mean and standard deviation are essential for constructing the confidence interval, as they provide the necessary statistics to estimate the range around the population mean.

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Related Practice

Textbook Question

In Exercises 3–6, construct the indicated confidence interval for the population mean . Which distribution did you use to create the confidence interval?

c=0.90, x̅=8.21, σ=0.62, n=8

Textbook Question

In Exercises 7–10, the statement represents a claim. Write its complement and state which is Ho and which is Ha.

σ=0.63

Textbook Question

In Exercises 3–6, construct the indicated confidence interval for the population mean . Which distribution did you use to create the confidence interval?

c=0.95, x̅=3.46, s=1.63, n=16

Textbook Question

The mean room rate for two adults for a random sample of 26 three-star hotels in Cincinnati has a sample standard deviation of \$31. Assume the population is normally distributed. (Adapted from Expedia)

Construct a 99% confidence interval for the population variance.

Textbook Question

Take this quiz as you would take a quiz in class. After you are done, check your work against the answers given in the back of the book.For each exercise, perform the steps below.

a. Identify the claim and state Ho and Ha

b. Determine whether the hypothesis test is left-tailed, right-tailed, or two-tailed, and whether to use a z-test or a t-test. Explain your reasoning.

c. Find the critical value(s) and identify the rejection region(s).

d. Find the appropriate standardized test statistic.

e. Decide whether to reject or fail to reject the null hypothesis.

f. Interpret the decision in the context of the original claim.

A music teacher claims that the mean scores on a music assessment test for eighth grade students in public and private schools are equal. The mean score for 13 randomly selected public school students is 146 with a standard deviation of 49, and the mean score for 15 randomly selected private school students is 160 with a standard deviation of 42. At α=0.1, can you reject the teacher’s claim? Assume the populations are normally distributed and the population variances are equal. (Adapted from National Center for Education Statistics)

Textbook Question

[APPLET] The annual earnings (in dollars) for 30 randomly selected locksmiths are shown below. Assume the population is normally distributed. (Adapted from Salary.com)

48,69446,85642,91261,67271,11254,861

69,45471,84159,75169,61254,28452,166

66,36048,16465,27235,25061,12765,397

58,92558,91659,01753,07045,19969,941

69,49257,08553,82952,69268,29853,792

A researcher claims that the mean annual earnings for locksmiths is \$55,000. At α=0.05, can you reject the researcher’s claim? Interpret the decision in the context of the original claim.