To test the hypothesis that the heights of women aged 25 to 34 have a different standard deviation than men aged 25 to 34, we will use the Chi-Squared test for the population variance.
Step-by-Step Solution:
1. State the Hypotheses:
- Null hypothesis ([tex]\(H_0\)[/tex]): The standard deviation of the heights of women aged 25 to 34 is equal to 2.9.
- Alternative hypothesis ([tex]\(H_1\)[/tex]): The standard deviation of the heights of women aged 25 to 34 is different from 2.9.
2. Significance Level:
- The significance level ([tex]\(\alpha\)[/tex]) is 0.05.
3. Sample Data:
- The heights of 16 randomly selected women aged 25 to 34 are:
62.1365, 63.8064, 64.2160, 64.1768, 64.2866, 64.4962, 64.1065, 64.7364,
64.8364, 65.2163, 65.1866, 65.7263, 65.0961, 65.1567, 65.5064, 65.65.
- Sample size ([tex]\(n\)[/tex]) is 16.
4. Calculate the Sample Standard Deviation:
- The sample standard deviation ([tex]\(s\)[/tex]) is 0.89035 (rounded to five decimal places).
5. Calculate the Chi-Squared Test Statistic:
- The test statistic is calculated by the formula:
[tex]\[
\chi^2 = \frac{(n-1) \times s^2}{\sigma^2}
\][/tex]
- Substituting the values:
[tex]\[
\chi^2 = \frac{(16-1) \times (0.89035)^2}{(2.9)^2} = 1.4139
\][/tex]
6. Determine the Critical Values:
- For a two-tailed test at [tex]\(\alpha = 0.05\)[/tex], the critical values for the Chi-Squared distribution with [tex]\(df = n - 1 = 15\)[/tex] degrees of freedom are:
- [tex]\(\chi^2_{left}\)[/tex] at [tex]\(\alpha/2 = 0.025\)[/tex] (lower critical value): 6.262
- [tex]\(\chi^2_{right}\)[/tex] at [tex]\(1 - \alpha/2 = 0.975\)[/tex] (upper critical value): 27.488
7. Decision Rule:
- If the test statistic lies outside the range defined by the critical values, we reject the null hypothesis. Otherwise, we do not reject the null hypothesis.
8. Compare the Test Statistic to the Critical Values:
- The test statistic is [tex]\(\chi^2 = 1.4139\)[/tex].
- The critical values are 6.262 and 27.488.
Since [tex]\(1.4139\)[/tex] is not within the range of the critical values [6.262, 27.488], the test statistic lies outside the critical region.
Thus, based on the calculations, we do not reject the null hypothesis. This means that there is not enough evidence at the 0.05 significance level to claim that the standard deviation of the heights of women aged 25 to 34 is different from 2.9.
Correct Response:
- Test statistic: [tex]\(\chi^2=1.4139\)[/tex]. Critical values: [tex]\(\chi^2=6.262, 27.488\)[/tex].
