Final answer:
To calculate the 95% confidence interval for yearly milk consumption, one must use the sample mean, standard deviation, sample size, and the t-distribution. For hypothesis testing, the null hypothesis that the mean is 80 gallons is tested against the alternative hypothesis using the t-test statistic and critical t-value for a 5% significance level.
Step-by-step explanation:
To calculate a 95% confidence interval for the mean yearly milk consumption, we will use the sample mean (\( \bar{x} \)), the standard deviation (s), and the sample size (n). Since the sample size is less than 30 and the population standard deviation is not known, we'll use the t-distribution.
Confidence Interval Calculation:
- Find the t-score that corresponds to a 95% confidence level for a sample size of 22 (degrees of freedom = n - 1 = 21). This value can be found in the t-distribution table or with a calculator (approximately 2.080).
- Calculate the standard error (SE) using the formula SE = s / \(\sqrt{n}\).
- Multiply the t-score by the SE to find the margin of error (MOE).
- Add and subtract the MOE from the sample mean to find the confidence interval.
Hypothesis Testing:
- State the null hypothesis H0: \( \mu = 80 \) gallons, and the alternative hypothesis Ha: \( \mu \\eq 80 \) gallons.
- Calculate the test statistic using the formula\( t = (\bar{x} - \mu) / (s / \(\sqrt{n}\)) \).
- Compare the test statistic to the critical t-value for a 5% significance level and 21 degrees of freedom (approximately \(\pm 2.080\)).
- If the test statistic is greater than the critical t-value in absolute terms, reject the null hypothesis; otherwise, do not reject the null hypothesis.
By following these steps, we can effectively estimate and test the hypothesis concerning the mean yearly milk consumption.