Annual per capita consumption of milk is 21.6 gallons (Statistical Abstract of the United States: 2006). Being from the Midwest, you believe milk consumption is higher there and wish to support your opinion. A sample of 16 individuals from the mid western town of Webster City showed a sample mean annual consumption of 24.1 gallons with a standard deviation of s = 4.8.
1. What is a point estimate of the difference between mean annual consumption in Webster City and the national mean?
2. At = .05, test for a significant difference by completing the following.
Calculate the value of the test statistic (to 2 decimals).
3. The p-value is?
1. To find the point estimate of the difference between mean annual consumption in Webster City and the national mean, subtract the national mean consumption (21.6 gallons) from the sample mean consumption in Webster City (24.1 gallons):
Point Estimate = Sample Mean - National Mean
Point Estimate = 24.1 - 21.6
Point Estimate = 2.5
Therefore, the point estimate of the difference between mean annual consumption in Webster City and the national mean is 2.5 gallons.
2. To test for a significant difference between mean annual consumption in Webster City and the national mean, we can use a t-test. The test statistic is calculated using the formula:
t = (Sample Mean - Population Mean) / (Standard Deviation / √Sample Size)
In this case, the sample mean annual consumption in Webster City is 24.1 gallons, the population mean is 21.6 gallons, the standard deviation is 4.8 gallons, and the sample size is 16.
t = (24.1 - 21.6) / (4.8 / √16)
t = 2.5 / (4.8 / 4)
t = 2.5 / 1.2
t = 2.08 (rounded to 2 decimal places)
Therefore, the value of the test statistic is 2.08.
3. The p-value is a measure of the strength of evidence against the null hypothesis. To find the p-value, we need to look up the corresponding probability in a t-distribution table or use statistical software.
Since the significance level (α) is given as 0.05, we compare the test statistic (t = 2.08) to the critical value(s) from the t-distribution at the corresponding degrees of freedom. The degrees of freedom for this test is n - 1 = 16 - 1 = 15.
If the calculated t-value is greater than the critical value or if the p-value is less than α (0.05), we can reject the null hypothesis and conclude that there is a significant difference between mean annual consumption in Webster City and the national mean.
However, without the degrees of freedom and the corresponding critical values, we cannot determine the exact p-value in this scenario.