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 calculate the point estimate of the difference between mean annual consumption in Webster City and the national mean, you can simply subtract the national mean from the sample mean.
Point estimate = Sample mean - National mean
= 24.1 gallons - 21.6 gallons
= 2.5 gallons
Therefore, the point estimate of the difference in mean annual consumption is 2.5 gallons.
2. To test for a significant difference between the mean annual consumption in Webster City and the national mean, you can perform a hypothesis test. Assuming that the population distribution is approximately normal and the sample size is small (n < 30), we can use a t-test.
The null hypothesis (H0) is that there is no significant difference between mean annual consumption in Webster City and the national mean.
The alternative hypothesis (H1) is that there is a significant difference between mean annual consumption in Webster City and the national mean.
Given that the sample size (n) is 16, the sample mean (x̄) is 24.1 gallons, and the sample standard deviation (s) is 4.8 gallons, we can calculate the test statistic using the formula:
t = (x̄ - μ) / (s / sqrt(n))
Where:
x̄ = Sample mean
μ = Population mean (national mean in this case)
s = Sample standard deviation
n = Sample size
t = (24.1 - 21.6) / (4.8 / sqrt(16))
= 2.5 / (4.8 / 4)
= 2.5 / 1.2
≈ 2.08 (rounded to 2 decimal places)
The calculated value of the test statistic is approximately 2.08.
3. To find the p-value associated with the calculated test statistic, we need to refer to the t-distribution table or use statistical software. Since the p-value is the probability of observing a test statistic as extreme as the calculated value, we can determine the p-value based on the degrees of freedom associated with the t-distribution.
Degrees of freedom (df) = Sample size - 1 = 16 - 1 = 15
At a significance level of 0.05 (α = 0.05), we need to determine the critical value of t from the t-distribution table with 15 degrees of freedom. Let's assume that the critical value is tα/2,df.
If the absolute value of the calculated t-statistic is greater than the critical value, then we reject the null hypothesis and conclude that there is a significant difference between mean annual consumption in Webster City and the national mean.
If the p-value associated with the calculated t-statistic is less than α (0.05 in this case), then we reject the null hypothesis and conclude that there is a significant difference between mean annual consumption in Webster City and the national mean.
To find the p-value, we can use statistical software or online calculators specific to hypothesis testing. The p-value associated with the calculated test statistic is the probability of obtaining a more extreme result (in favor of the alternative hypothesis) assuming the null hypothesis is true.
Unfortunately, I can't provide the exact p-value without further information. However, you can consult statistical software or reference t-distribution tables to find the p-value corresponding to the calculated test statistic.