Mail security estimates that the average daily per-store theft is less than $250, but wants to determine accuracy of this statistic. The company researcher takes a sample of 81 clerks and finds that X =$260 and s=$20.
a) Test at a=.05
b) Construct a 95% CIE of u
To test the accuracy of the statistic, we can perform a hypothesis test and construct a confidence interval. Let's go step by step:
a) Testing at a = 0.05:
We need to set up the null and alternative hypotheses and perform the hypothesis test.
Null hypothesis (H0): The average daily per-store theft is less than or equal to $250.
Alternative hypothesis (Ha): The average daily per-store theft is greater than $250.
To conduct the hypothesis test, we will use a one-sample t-test, since we have a sample mean and standard deviation. The test statistic can be calculated using the formula:
t = (X - μ) / (s / √n)
where X is the sample mean, μ is the population mean (hypothesized value under the null hypothesis), s is the sample standard deviation, and n is the sample size.
Given the sample values:
X = $260
s = $20
n = 81
Let's calculate the test statistic:
t = (260 - 250) / (20 / √81)
t = 10 / (20 / 9)
t = 4.5
Now we need to compare the test statistic with the critical value from the t-distribution. Since we set a = 0.05, which corresponds to a 95% confidence level, we will find the critical value with (1 - a) = 0.95 and degrees of freedom (df) = n - 1 = 81 - 1 = 80.
Looking up the critical value in a t-table or using statistical software, we find the critical value to be approximately 1.663 (rounded to three decimal places).
Since our test statistic t (4.5) is greater than the critical value (1.663), we reject the null hypothesis. This means there is sufficient evidence to conclude that the average daily per-store theft is greater than $250.
b) Constructing a 95% Confidence Interval Estimate (CIE) for μ:
To construct the confidence interval estimate for the population mean, we can use the formula:
CIE = X ± (t * s / √n)
where X is the sample mean, t is the critical value from the t-distribution corresponding to the desired confidence level and degrees of freedom, s is the sample standard deviation, and n is the sample size.
Using the same values as before:
X = $260
s = $20
n = 81
Looking up the critical value with a 95% confidence level and 80 degrees of freedom, we find it to be approximately 1.990 (rounded to three decimal places).
Now, let's calculate the confidence interval:
CIE = 260 ± (1.990 * 20 / √81)
CIE = 260 ± (1.990 * 20 / 9)
CIE = 260 ± 4.414
The 95% confidence interval estimate for the population mean (μ) is $260 ± $4.414, which can be written as ($255.586, $264.414).
This means we are 95% confident that the true average daily per-store theft falls within this interval.