how can I determine weather to reject or fail to reject the following nul hypothosis: let a= o.01, data: s= 3.1, n=55 and H(0) st dev 2.8 and H(1)< 2.8

To determine whether to reject or fail to reject the null hypothesis, you need to conduct a hypothesis test using the given data and significance level. Here's how you can proceed:

1. Define the null hypothesis (H₀) and the alternative hypothesis (H₁):
- Null hypothesis (H₀): The population standard deviation is equal to or greater than 2.8 (σ ≥ 2.8).
- Alternative hypothesis (H₁): The population standard deviation is less than 2.8 (σ < 2.8).

2. Determine the test statistic:
- For this case, since you have the sample standard deviation (s), you can use the chi-square test statistic:

chi-square = (n - 1) * s^2 / σ₀^2

where n is the sample size, s is the sample standard deviation, and σ₀ is the hypothesized population standard deviation under the null hypothesis.

- Plugging in the given values, you get:

chi-square = (55 - 1) * 3.1^2 / 2.8^2

3. Determine the critical value or p-value:
- Since the alternative hypothesis states that the population standard deviation is less than 2.8, this is a left-tailed test.
- Depending on the level of significance (α), you can find the critical value from the chi-square distribution table or use statistical software.
- Considering α = 0.01, the critical value would be obtained from the chi-square distribution table with 54 degrees of freedom. For α = 0.01, the critical value is approximately 26.509.

4. Compare the test statistic and the critical value:
- If the test statistic is less than the critical value, you reject the null hypothesis.
- If the test statistic is greater than or equal to the critical value, you fail to reject the null hypothesis.

5. Conclusion:
- If the test statistic (chi-square) is less than the critical value of 26.509, you would reject the null hypothesis and conclude that there is evidence to suggest that the population standard deviation is less than 2.8.
- If the test statistic is greater than or equal to the critical value, you would fail to reject the null hypothesis and conclude that there is not enough evidence to suggest that the population standard deviation is less than 2.8.

You now have the necessary information to determine whether to reject or fail to reject the null hypothesis based on the given data.