Becky owns a diner and is concerned about sustaining the business.She wants to ascertain if the standard deviation of the profits for each week is greater than $250.The details of the profits for the week are (in dollars) 1,743 1,438 1,212 1,705 1,985 1,857 1,916 Assume that profits are normally distributed.Using the critical value approach at α = 0.05,which of the following is the correct conclusion for Becky's concern? A) We reject H₀becausethe value of the test statistic is greater than . B) We do not reject H₀becausethe value of the test statistic is greater than . C) We reject H₀becausethe value of the test statistic is less than D) We do not reject H0becausethe value of the test statistic is less than

Question

Quizplus · Accepted Answer

We can use the formula for the test statistic for a one-sample t-test:

t = (x̄ - μ) / (s / √n)

where x̄ is the sample mean, μ is the hypothesized population mean (which is not given but is not needed for this test), s is the sample standard deviation, and n is the sample size.

First, we need to calculate the sample standard deviation:

s = √[Σ(xi - x̄)2 / (n - 1)]
  = √[(1743 - 1704.28)2 + (1438 - 1704.28)2 + (1212 - 1704.28)2 + (1705 - 1704.28)2 + (1985 - 1704.28)2 + (1857 - 1704.28)2 + (1916 - 1704.28)2 / 6]
  ≈ 290.47

Next, we calculate the test statistic:

t = (x̄ - μ) / (s / √n)
  = (1704.28 - μ) / (290.47 / √7)

From the question stem, we have α = 0.05 and degrees of freedom (df) = n - 1 = 6. Using a t-distribution table or calculator with df = 6 and alpha = 0.05, we find the critical values to be approximately ±2.447.

Since we are testing if the standard deviation is greater than $250, this is a one-tailed test. Therefore, we only need to check if the test statistic is greater than the one-tailed critical value.

If we multiply the critical value by the standard error, we obtain the critical value for the test statistic:

Critical value = t-critical * (s / √n)
              = 2.447 * (290.47 / √7)
              ≈ 288.95

The test statistic is:

t = (1704.28 - μ) / (290.47 / √7)

Since we are looking at a one-tailed test for a larger standard deviation, we will use the right-tailed test. The reject region is: t > 288.95

The computed test statistic t=0.66 is not in the rejection region. Therefore, we do not reject H0 stating that the standard deviation is less than or equal to $250. This means that there is not enough evidence to conclude that the standard deviation of weekly profits is greater than $250 for the diner.

Becky Owns a Diner and Is Concerned About Sustaining the Business.She