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 the mean annual consumption in Webster City and the national mean, we need to subtract the national mean from the sample mean of Webster City.
Point estimate = Sample mean - National mean
Point estimate = 24.1 - 21.6
Point estimate = 2.5 gallons
So, the point estimate of the difference is 2.5 gallons.
2. To test for a significant difference between the mean annual consumption in Webster City and the national mean, we can use a t-test. The formula for the t-test statistic is:
t = (Sample mean - Population mean) / (Sample standard deviation / sqrt(sample size))
In this case:
Sample mean = 24.1 gallons
Population mean = 21.6 gallons
Sample standard deviation = 4.8 gallons
Sample size = 16 individuals
Plugging in the values:
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 at the given significance level of α = 0.05.
Since we don't have the exact distribution of the sample mean, we'll use the t-distribution. With a sample size of 16, degrees of freedom (df) = 16 - 1 = 15.
Using a t-table or a statistical software, we can find the critical value for a two-tailed test with α = 0.05 and df = 15. Let's assume the critical value is 2.131.
If the absolute value of the test statistic is greater than the critical value, then we reject the null hypothesis.
In this case, |2.08| < 2.131, so we do not reject the null hypothesis.
The p-value is the probability of obtaining a test statistic as extreme as (or more extreme than) the observed value, assuming the null hypothesis is true. Since we do not reject the null hypothesis, the p-value would be greater than 0.05.
Therefore, without knowing the exact p-value, we can conclude that the p-value is greater than 0.05.