Math  /  Data & Statistics

QuestionSeveral years ago, the mean height of women 20 years of age or older was 63.7 inches. Suppose that a random sample of 45 women who are 20 years of age or older today results in a mean height of 64.3 inches. (a) State the appropriate null and alternative hypotheses to assess whether women are taller today. (b) Suppose the P -value for this test is 0.13 . Explain what this value represents. (c) Write a conclusion for this hypothesis test assuming an α=0.10\alpha=0.10 level of significance. (a) State the appropriate null and alternative hypotheses to assess whether women are taller today. A. H0:μ=64.3H_{0}: \mu=64.3 in. versus H1:μ>64.3H_{1}: \mu>64.3 in. B. H0:μ=63.7H_{0}: \mu=63.7 in. versus H1:μ63.7inH_{1}: \mu \neq 63.7 \mathrm{in}. C. H0:μ=63.7H_{0}: \mu=63.7 in. versus H1:μ>63.7H_{1}: \mu>63.7 in. D. H0:μ=64.3H_{0}: \mu=64.3 in. versus H1:μ64.3inH_{1}: \mu \neq 64.3 \mathrm{in}. E. H0:μ=63.7H_{0}: \mu=63.7 in. versus H1:μ<63.7inH_{1}: \mu<63.7 \mathrm{in}. F. H0:μ=64.3H_{0}: \mu=64.3 in. versus H1:μ<64.3H_{1}: \mu<64.3 in.

Studdy Solution

STEP 1

1. The population mean height several years ago was 63.7 inches.
2. The sample mean height today is 64.3 inches.
3. We are conducting a hypothesis test to determine if women are taller today.
4. The level of significance for the test is α=0.10\alpha = 0.10.

STEP 2

1. Identify the null and alternative hypotheses.
2. Interpret the P-value.
3. Draw a conclusion based on the P-value and the level of significance.

STEP 3

Identify the null and alternative hypotheses.
- The null hypothesis (H0H_0) represents the status quo or the claim to be tested. In this case, it is that the mean height of women today is the same as several years ago, 63.7 inches. - The alternative hypothesis (H1H_1) represents the claim we are testing for, which is that women are taller today.
The appropriate hypotheses are:
H0:μ=63.7 inches H_0: \mu = 63.7 \text{ inches} H1:μ>63.7 inches H_1: \mu > 63.7 \text{ inches}
This corresponds to option C.

STEP 4

Interpret the P-value.
- The P-value is the probability of observing a sample mean as extreme as 64.3 inches, or more extreme, assuming the null hypothesis is true. - A P-value of 0.13 indicates there is a 13% chance of observing a sample mean of 64.3 inches or greater if the true population mean is 63.7 inches.

STEP 5

Draw a conclusion based on the P-value and the level of significance.
- Compare the P-value to the level of significance (α=0.10\alpha = 0.10). - Since the P-value (0.13) is greater than α\alpha (0.10), we fail to reject the null hypothesis.
Conclusion: There is not enough statistical evidence at the α=0.10\alpha = 0.10 level of significance to conclude that women are taller today than several years ago.

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