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 from the sample mean.
Point estimate = Sample mean - National mean
Point estimate = 24.1 - 21.6
Point estimate = 2.5 gallons
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 at a significance level of 0.05, we need to calculate the test statistic.
The formula for the test statistic when comparing two means is:
t = (sample mean - hypothesized mean) / (standard deviation / sqrt(sample size))
Given values:
Sample mean (x̄) = 24.1 gallons
Hypothesized mean (μ) = 21.6 gallons
Standard deviation (s) = 4.8 gallons
Sample size (n) = 16
We can substitute these values into the formula to calculate the test statistic:
t = (24.1 - 21.6) / (4.8 / sqrt(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 approximately 2.08.
3. To find the p-value, we need to compare the test statistic to the critical value from the t-distribution table.
The p-value is the probability of observing a test statistic as extreme or more extreme than the one calculated under the null hypothesis (i.e., no difference between the means).
Using the test statistic (t = 2.08) and the degrees of freedom (n - 1 = 16 - 1 = 15), we can look up the p-value in the t-distribution table or use statistical software to determine it.
Assuming a two-tailed test (since we are testing for a difference, not a specific direction), the p-value for t = 2.08 and degrees of freedom = 15 is approximately 0.051 (rounded to 3 decimal places).
Therefore, the p-value is approximately 0.051.
Note: The p-value of 0.051 is slightly larger than the significance level of 0.05. If the p-value is less than the significance level, we would reject the null hypothesis. However, in this case, the p-value is greater than the significance level, so we would fail to reject the null hypothesis and conclude that there is not enough evidence to support a significant difference between the mean annual consumption in Webster City and the national mean.