Math  /  Data & Statistics

Question4. In the United States, it is well-known that for the population of males, the mean height is 69.1 inches with standard deviation 4 inches. A researcher believes that a certain group is significantly taller than other males, due to genetic factors. For a sample of 25 such men, the mean is 71.2 inches with standard deviation 3.5 inches. a. Can it be reasonably assumed that the population is normally distributed? Are the criteria met to perform a z-test for the mean? Explain. b. Set up the null and alternative hypothesis to test if this population is taller than average. c. Calculate the value of the test statistic. d. Find the p-value. e. At the 5%5 \% level of significance (α=0.05)(\alpha=0.05), make a decision whether to reject H0H_{0}. f. What can we conclude?

Studdy Solution

STEP 1

1. The sample size is n=25 n = 25 .
2. The population mean height for males is μ=69.1 \mu = 69.1 inches.
3. The population standard deviation is σ=4 \sigma = 4 inches.
4. The sample mean height is xˉ=71.2 \bar{x} = 71.2 inches.
5. The sample standard deviation is s=3.5 s = 3.5 inches.
6. The level of significance is α=0.05 \alpha = 0.05 .

STEP 2

1. Assess normality and criteria for a z-test.
2. Formulate the null and alternative hypotheses.
3. Calculate the test statistic.
4. Determine the p-value.
5. Make a decision based on the p-value.
6. Draw a conclusion from the hypothesis test.

STEP 3

Assess whether the population can be assumed to be normally distributed and if the criteria for a z-test are met.
- The Central Limit Theorem states that for a sample size n30 n \geq 30 , the sampling distribution of the sample mean is approximately normal, regardless of the population distribution. However, since n=25 n = 25 , we rely on the assumption that the population is normally distributed. - The sample size is less than 30, but since we are given the population standard deviation (σ=4 \sigma = 4 ), we can use a z-test.

STEP 4

Set up the null and alternative hypotheses.
- Null Hypothesis (H0 H_0 ): The mean height of the group is equal to the population mean, μ=69.1 \mu = 69.1 . - Alternative Hypothesis (Ha H_a ): The mean height of the group is greater than the population mean, μ>69.1 \mu > 69.1 .

STEP 5

Calculate the test statistic using the formula for the z-test:
z=xˉμσnz = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}}
Substitute the given values:
z=71.269.1425=2.10.8=2.625z = \frac{71.2 - 69.1}{\frac{4}{\sqrt{25}}} = \frac{2.1}{0.8} = 2.625

STEP 6

Find the p-value for the calculated z-test statistic.
- Since this is a one-tailed test (greater than), we find the p-value corresponding to z=2.625 z = 2.625 .
Using a standard normal distribution table or calculator, the p-value is approximately 0.0043 0.0043 .

STEP 7

Make a decision based on the p-value.
- Compare the p-value to the significance level α=0.05 \alpha = 0.05 . - Since 0.0043<0.05 0.0043 < 0.05 , we reject the null hypothesis H0 H_0 .

STEP 8

Conclude the hypothesis test.
- Since we rejected H0 H_0 , we conclude that there is significant evidence at the 5% 5\% level of significance to support the claim that the group is taller than the average male population.

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