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. The point estimate of the difference between the mean annual consumption in Webster City and the national mean is the sample mean minus the population mean. Therefore, the point estimate is 24.1 - 21.6 = 2.5 gallons.
2. To test for a significant difference, we can use a t-test. The formula for the test statistic is given by:
t = (sample mean - population mean) / (sample standard deviation / √n)
Plugging in the given values:
t = (24.1 - 21.6) / (4.8 / √16)
t = 2.5 / (4.8 / 4)
t ≈ 2.5 / 1.2
t ≈ 2.08
The value of the test statistic is approximately 2.08 (to 2 decimal places).
3. To determine the p-value, we would need the degrees of freedom (df). Since we have a sample size of 16, the degrees of freedom can be calculated as df = n - 1 = 16 - 1 = 15.
Using the t-distribution table or statistical software, we can find the p-value associated with the test statistic of t = 2.08 and df = 15. The p-value for this test is the probability of obtaining a test statistic as extreme as 2.08 or more extreme in the direction of the alternative hypothesis.
The p-value associated with t = 2.08 and df = 15 is approximately 0.053 (to 3 decimal places).
1. To find the point estimate of the difference between the mean annual consumption in Webster City and the national mean, you need to 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
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, you can perform a hypothesis test.
Null Hypothesis (H0): There is no significant difference between the mean annual consumption in Webster City and the national mean.
Alternative Hypothesis (Ha): There is a significant difference between the mean annual consumption in Webster City and the national mean.
Since the sample size is small (n=16) and the population standard deviation (σ) is unknown, you can use the t-distribution to calculate the test statistic.
The formula for the t-test statistic is:
t = (x̄ - μ) / (s / √n)
where x̄ is the sample mean, μ is the population mean (in this case, the national mean), s is the sample standard deviation, and n is the sample size.
Plugging in the values:
x̄ = 24.1
μ = 21.6
s = 4.8
n = 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)
The test statistic value is approximately 2.08.
3. To find the p-value, you need to compare the test statistic value to the t-distribution table or use statistical software.
Since the significance level (α) is given as 0.05, you are testing at a 95% confidence level. This means that you are looking for a two-tailed test and need to find the critical t-value that corresponds to an alpha level of 0.025 (0.05 / 2) in each tail.
Using the t-distribution table or statistical software, you can find the critical t-value. Let's assume it is t_critical = 2.145 for a degree of freedom (df) of 15 (n-1).
If the absolute value of the test statistic value (|t|) is greater than the critical t-value (|t_critical|), then the p-value is less than the significance level (α), and you reject the null hypothesis. Otherwise, you fail to reject the null hypothesis.
In this case, |t| = 2.08, which is less than |t_critical| = 2.145. Therefore, the p-value is greater than 0.05, and we fail to reject the null hypothesis.
Unfortunately, without the exact distribution of the sample, we cannot determine the exact p-value. However, since the p-value is greater than 0.05, it suggests that there is not enough evidence to support a significant difference in milk consumption between Webster City and the national mean at the 95% confidence level.