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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
All textbooksLarson 8th EditionCh. 8 - Hypothesis Testing with Two SamplesProblem 8.6-8-11
Chapter 8, Problem 8.6-8-11

A pediatrician claims that the mean birth weight of a single-birth baby is greater than the mean birth weight of a baby that has a twin. The mean birth weight of a random sample of 85 single-birth babies is 3086 grams. Assume the population standard deviation is 563 grams. The mean birth weight of a random sample of 68 babies that have a twin is 2263 grams. Assume the population standard deviation is 624 grams. At α=0.10, can you support the pediatrician’s claim? Interpret the decision in the context of the original claim.

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Step 1: Define the null hypothesis (H₀) and the alternative hypothesis (H₁). The null hypothesis is H₀: μ₁ ≤ μ₂, where μ₁ is the mean birth weight of single-birth babies and μ₂ is the mean birth weight of twin babies. The alternative hypothesis is H₁: μ₁ > μ₂, which aligns with the pediatrician's claim.
Step 2: Identify the test statistic to use. Since the population standard deviations are known, use the z-test for the difference between two means. The formula for the z-test statistic is: (x1-x2)-0σ12n1+σ22n2, where x1 and x2 are the sample means, σ1 and σ2 are the population standard deviations, and n1 and n2 are the sample sizes.
Step 3: Plug in the given values into the formula. Here, x1 = 3086, x2 = 2263, σ1 = 563, σ2 = 624, n1 = 85, and n2 = 68. Calculate the numerator (x1 - x2) and the denominator (the square root of the sum of variances divided by their respective sample sizes).
Step 4: Determine the critical value for a one-tailed z-test at α = 0.10. Use a z-table or statistical software to find the critical z-value corresponding to a significance level of 0.10. This value will be compared to the calculated z-test statistic.
Step 5: Compare the calculated z-test statistic to the critical z-value. If the z-test statistic is greater than the critical z-value, reject the null hypothesis (H₀) and conclude that there is sufficient evidence to support the pediatrician's claim. Otherwise, fail to reject the null hypothesis. Interpret the result in the context of the problem, explaining whether the data supports the claim that single-birth babies have a greater mean birth weight than twin babies.

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Key Concepts

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

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about population parameters based on sample data. In this context, the pediatrician's claim serves as the alternative hypothesis, while the null hypothesis posits that there is no difference in mean birth weights between single-birth and twin babies. The process involves calculating a test statistic and comparing it to a critical value to determine whether to reject or fail to reject the null hypothesis.
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Step 1: Write Hypotheses

Confidence Intervals

A confidence interval provides a range of values within which we can expect a population parameter to lie, based on sample data. In this scenario, constructing confidence intervals for the mean birth weights of both groups can help assess the overlap between them. If the intervals do not overlap, it may provide evidence supporting the pediatrician's claim that single-birth babies have a higher mean birth weight than twins.
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Introduction to Confidence Intervals

P-Value

The p-value is a statistical measure that helps determine the significance of the results obtained from hypothesis testing. It represents the probability of observing the sample data, or something more extreme, if the null hypothesis is true. In this case, calculating the p-value will help assess whether the observed difference in mean birth weights is statistically significant at the α=0.10 level, thus supporting or refuting the pediatrician's claim.
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Step 3: Get P-Value
Related Practice
Textbook Question

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


μ≠2.28

Textbook Question

Constructing Confidence Intervals for μ1-μ2, When the sampling distribution for x̅1-x̅2 is approximated by a t-distribution and the population variances are not equal, you can construct a confidence interval for μ1-μ2 , as shown below.

construct the indicated confidence interval for μ1-μ2 . Assume the populations are approximately normal with unequal variances.

10K Race

To compare the mean finishing times of male and female participants in a 10K race, you randomly select several finishing times from both sexes. The results are shown at the left. Construct an 80% confidence interval for the difference in mean finishing times of male and female participants in the race. (Adapted from Great Race)


Textbook Question

Test the claim about the mean of the differences for a population of paired data at the level of significance α. Assume the samples are random and dependent, and the populations are normally distributed.

Claim: μd≤0 , α=0.10, Sample statistics: d̄ =6.5, sd=9.54, n=16

Textbook Question

Gas Mileage The table shows the gas mileages (in miles per gallon) of eight cars with and without using a fuel additive. At α=0.10, is there enough evidence to conclude that the additive improved gas mileage? Assume the populations are normally distributed.


Textbook Question

Describe another way you can perform a hypothesis test for the difference between the means of two populations using independent samples with and known that does not use rejection regions.

Textbook Question

Independent and Dependent Samples In Exercises 5–8, classify the two samples as independent or dependent and justify your answer.

Sample 1: The commute times of 10 workers when they use their own vehicles

Sample 2: The commute times of the same 10 workers when they use public transportation

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