Test the hypothesis that American females are taller than their British counterparts at α=5%. A random sample of size 25 showed a mean of 65 inches with a standard deviation of 25... Test the hypothesis that American females are taller than their British counterparts at α=5%. A random sample of size 25 showed a mean of 65 inches with a standard deviation of 25 inches. a. Determine 99% confidence interval for the mean of the population. b. Test significant or not at 5% level of significance. Read more

Steps to Solve

1. Identify the Hypotheses

   We need to set up the null and alternative hypotheses.

   • Null hypothesis ($H_0$): The average height of American females is less than or equal to that of British females.
   • Alternative hypothesis ($H_1$): The average height of American females is greater than that of British females.

2. Collect the Data

   From the information provided:

   • Average height of American females ($\bar{X}_1$) = 65.2 inches
   • Average height of British females ($\bar{X}_2$) = 64.5 inches
   • Standard deviation of American females ($s_1$) = 2.5 inches
   • Standard deviation of British females ($s_2$) = 2.8 inches
   • Sample sizes ($n_1 = 50$, $n_2 = 50$)

3. Calculate the Test Statistic

   We will use a t-test for this hypothesis test due to the sample sizes and the standard deviations. The t-statistic is calculated using the formula:

   $$ t = \frac{(\bar{X}_1 - \bar{X}_2)}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}} $$

   Substituting the values:

   $$ t = \frac{(65.2 - 64.5)}{\sqrt{\frac{2.5^2}{50} + \frac{2.8^2}{50}}} $$

4. Calculate the Degrees of Freedom

   Assuming unequal variances, we can use the formula for degrees of freedom:

   $$ df = \frac{\left(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}\right)^2}{\frac{\left(\frac{s_1^2}{n_1}\right)^2}{n_1 - 1} + \frac{\left(\frac{s_2^2}{n_2}\right)^2}{n_2 - 1}} $$

5. Determine the Critical Value

   Using a t-distribution table, we look up the critical t-value for a one-tailed test at $\alpha = 0.05$ with the calculated degrees of freedom.

6. Make a Decision

   If the calculated t-value is greater than the critical value, we reject the null hypothesis.