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, you can 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, we need to conduct a hypothesis test. The null hypothesis (H₀) states that there is no significant difference between the mean annual consumption in Webster City and the national mean. The alternative hypothesis (H₁) states that there is a significant difference between the two means.
H₀: Mean annual consumption in Webster City - National mean = 0
H₁: Mean annual consumption in Webster City - National mean ≠ 0
Next, we calculate the test statistic using the formula:
Test statistic = (Sample mean - National mean) / (Standard deviation / sqrt(sample size))
Test statistic = (24.1 - 21.6) / (4.8 / sqrt(16))
Test statistic = 2.5 / (4.8 / 4)
Test statistic ≈ 2.0833 (rounded to 2 decimal places)
3. To find the p-value for the test statistic, we compare it to the critical value associated with the significance level (α). Since α is given as 0.05 (or 5%), we will use this value to determine the p-value.
The test statistic follows a t-distribution with (n - 1) degrees of freedom. In this case, the sample size is 16, so the degrees of freedom is 15. By referring to a t-distribution table or using statistical software, we find that the critical t-value for a two-tailed test at α = 0.05 is approximately ±2.131.
Since the test statistic (2.0833) is not in the critical region beyond ±2.131, we fail to reject the null hypothesis. In other words, there is not enough evidence to conclude a significant difference between the mean annual consumption in Webster City and the national mean.
Therefore, the p-value is greater than 0.05, but the exact value is not provided in the question.