Ch. 8 - Hypothesis Testing with Two Samples

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 8 - Hypothesis Testing with Two Samples

Problem 8.1.29

Chapter 8, Problem 8.1.29

Constructing Confidence Intervals for μ1-μ2. You can construct a confidence interval for the difference between two population means μ1-μ2 , as shown below, when both population standard deviations are known, and either both populations are normally distributed or both n1>= 30 and n2>=30 . Also, the samples must be randomly selected and independent.
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In Exercises 29 and 30, construct the indicated confidence interval for μ1-μ2 .

Software Engineer Salaries Construct a 95% confidence interval for the difference between the mean annual salaries of entry level software engineers in Santa Clara, California, and Greenwich, CT, using the data from Exercise 27.

Verified step by step guidance

Step 1: Identify the given data from the problem. You need the sample means (x̄₁ and x̄₂), population standard deviations (σ₁ and σ₂), sample sizes (n₁ and n₂), and the confidence level (95%).

Step 2: Determine the critical value (z*) for a 95% confidence level. Since the population standard deviations are known, use the standard normal distribution (Z-distribution) to find the z* value corresponding to a 95% confidence level.

Step 3: Calculate the standard error (SE) for the difference between the two means using the formula:

\sqrt{\frac{σ^{1}}{2} + \frac{σ^{2}}{2}}

, where σ₁ and σ₂ are the population standard deviations, and n₁ and n₂ are the sample sizes.

Step 4: Compute the margin of error (ME) using the formula:

z * SE

, where z* is the critical value and SE is the standard error calculated in Step 3.

Step 5: Construct the confidence interval for μ₁ - μ₂ using the formula:

(x̄ 1 - x̄ 2) \pm ME

, where x̄₁ and x̄₂ are the sample means, and ME is the margin of error. This will give you the lower and upper bounds of the confidence interval.

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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 true population parameter. It is expressed with a certain level of confidence, such as 95%, indicating the probability that the interval includes the parameter. For example, if a 95% confidence interval for the difference between two means is calculated, it suggests that if the same sampling process were repeated multiple times, approximately 95% of the intervals would contain the true difference.

Independent Samples

Independent samples refer to two or more groups of data that are collected separately and do not influence each other. In the context of constructing confidence intervals for the difference between two means, it is crucial that the samples are independent to ensure that the results are valid. For instance, comparing salaries of software engineers from two different cities assumes that the selection of engineers in one city does not affect the selection in the other.

Normal Distribution and Sample Size

The assumption of normal distribution is important when constructing confidence intervals, especially when sample sizes are small. If the populations are normally distributed, the sample means will also be normally distributed. However, if the sample sizes are large (n1 ≥ 30 and n2 ≥ 30), the Central Limit Theorem states that the sampling distribution of the mean will approximate normality regardless of the population distribution, allowing for valid confidence interval construction.

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Sampling Distribution of Sample Proportion

Related Practice

Textbook Question

What conditions are necessary in order to use the z-test to test the difference between two population means?

Textbook Question

Test the claim about the difference between two population means and at the level of significance α. Assume the samples are random and independent, and the populations are normally distributed.

Claim: μ1=μ2, α=0.01, Assume (σ1)^2=(σ2)^2

Textbook Question

[APPLET] Teaching Methods

A new method of teaching reading is being tested on third grade students. A group of third grade students is taught using the new curriculum. A control group of third grade students is taught using the old curriculum. The reading test scores for the two groups are shown in the back-to-back stem-and-leaf plot.

At , α=0.10 is there enough evidence to support the claim that the new method of teaching reading produces higher reading test scores than the old method does? Assume the population variances are equal.

Textbook Question

Blue Crabs A marine researcher claims that the stomachs of blue crabs from one location contain more fish than the stomachs of blue crabs from another location. The stomach contents of a sample of 25 blue crabs from Location A contain a mean of 320 milligrams of fish and a standard deviation of 60 milligrams. The stomach contents of a sample of 15 blue crabs from Location B contain a mean of 280 milligrams of fish and a standard deviation of 80 milligrams. At , α= 0.01can you support the marine researcher’s claim? Assume the population variances are equal.

Textbook Question

Testing the Difference Between Two Means In Exercises 15–24, (a) identify the claim and state Ho and Ha, (b) find the critical value(s) and identify the rejection region(s), (c) find the standardized test statistic z, (d) decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. Assume the samples are random and independent, and the populations are normally distributed.

Braking Distances To compare the dry braking distances from 60 to 0 miles per hour for two makes of automobiles, a safety engineer conducts braking tests for 16 compact SUVs and 11 midsize SUVs. The mean braking distance for the compact SUVs is 131.8 feet. Assume the population standard deviation is 5.5 feet. The mean braking distance for the midsize SUVs is 132.8 feet. Assume the population standard deviation is 6.7 feet. At α=0.10 , can the engineer support the claim that the mean braking distances are different for the two categories of SUVs? (Adapted from Consumer Reports)

Textbook Question

In Exercises 11–14, test the claim about the difference between two population means and at the level of significance . Assume the samples are random and independent, and the populations are normally distributed.

Claim: μ1<μ2; α=0.03

Population statistics:σ1=136 and σ2=215

Sample Statistics: x̅1=5004, n1=144, x̅2=4895, n2=156