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 will perform a hypothesis test using the given sample data.
a) Hypothesis Test:
In this case, we want to test whether the average daily per-store theft is significantly different from $250. Our null hypothesis (H0) is that the average daily per-store theft is equal to $250, and the alternate hypothesis (Ha) is that the average daily per-store theft is not equal to $250.
H0: μ = $250 (null hypothesis)
Ha: μ ≠ $250 (alternate hypothesis)
To conduct the hypothesis test, we will use the t-test since we have a sample size smaller than 30 and the population standard deviation is unknown.
The test statistic formula for this case is:
t = (X - μ) / (s / √n)
where X is the sample mean, μ is the population mean, s is the sample standard deviation, and n is the sample size.
Given:
X = $260
s = $20
n = 81
Plugging in these values into the formula, we get:
t = ($260 - $250) / ($20 / √81)
t = $10 / ($20 / 9)
t = $10 * (9 / $20)
t ≈ 4.5
Next, we need to determine the critical value for our test at a significance level of α = 0.05. Since this is a two-tailed test, we will divide the significance level by 2 to find each tail's cutoff.
Using a t-table or t-distribution calculator with degrees of freedom (df) = n - 1 = 81 - 1 = 80, we find that the critical value for a two-tailed test at α = 0.025 is approximately ±1.992.
Since our computed t-value of 4.5 is outside the critical value range of ±1.992, we reject the null hypothesis (H0) and conclude that there is evidence to suggest that the average daily per-store theft is significantly different from $250.
b) Confidence Interval Estimation (CIE):
To construct a confidence interval estimate of the population mean, we will use the given sample data.
Given:
X = $260
s = $20
n = 81
Using the formula for a confidence interval estimate of the mean:
CIE = X ± t * (s / √n)
where X is the sample mean, t is the critical value based on the desired confidence level and degrees of freedom, s is the sample standard deviation, and n is the sample size.
For a 95% confidence level, we find the critical value from the t-table or t-distribution calculator with df = n - 1 = 81 - 1 = 80. The critical value for a 95% confidence interval is approximately ±1.990.
Plugging in the values, we get:
CIE = $260 ± 1.990 * ($20 / √81)
CIE = $260 ± 1.990 * ($20 / 9)
CIE ≈ $260 ± $8.85
Therefore, the 95% confidence interval estimate of the population mean is approximately $251.15 to $268.85.