Thorndike Sports Equipment has submitted a bid to be the sole supplier of swimmi

Question

Thorndike Sports Equipment has submitted a bid to be the sole supplier of swimming goggles for the U.S. Olympic Team. OptiView, Inc., has been supplying the goggles for many years, and the Olympic Committee has said it will switch to Thorndike only if the Thorndike goggles are found to be markedly better. They will use a standard leakage test, to test the goggles.

For purposes of fairness, the committee has purchased 16 examples from each manufacturer in the retail marketplace. This is to avoid the possibility that either manufacturer might supply goggles that have been specially modified for the test. Testing involves installing the goggles on a surface that simulates the face of a swimmer, then submitting them to increasing water pressure (expressed in meters of water depth) until the goggles experience leakage. The greater the number of meters before leakage, the better the quality of the goggles.

Both companies have received copies of the test results and have an opportunity to offer their respective comments before the final decision is made. The data are given below.

1. Based on analysis of these data, formulate a commentary that Ted Thorndike might wish to make to the committee and PREPARE A SHORT WRITTEN REPORT on why your company should get the contract.

Sixteen Thorndike Goggles 72 117 91 85 120 71 101 118 106 114 106 95 101 92 94 118 Sixteen OptiView Goggles 83 95 83 96 80 103 86 100 92 108 94 77 99 90 97 83

Explanation / Answer

We will be conducting a hypothesis test to the performance of Thorndike Goggles over OptiView Goggles.

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis: The mean water depth before leakage of Thorndike Goggles is equal to that of OptiView Goggles. That is 1 - 2 = 0
Alternative hypothesis: The mean water depth before leakage of Thorndike Goggles is greater than that of OptiView Goggles. That is 1 - 2 > 0

Formulate an analysis plan. For this analysis, we assume the significance level is 0.05. Using sample data, we will conduct a two-sample t-test of the null hypothesis.

Analyze sample data. Using sample data, we compute the standard error (SE), degrees of freedom (DF), and the t statistic test statistic (t).

Mean water depth before leakage of Thorndike Goggles = 100.063

Mean water depth before leakage of OptiView Goggles = 91.625

Standard deviation of water depth before leakage of Thorndike Goggles, s1 = 15.627

Standard deviation of water depth before leakage of OptiView Goggles, s2 = 8.921

Let s1 is the  standard deviation of sample 1, s2 is the standard deviation of sample 2, n1 is the size of sample 1, n2 is the size of sample 2, x1 is the mean of sample 1, x2 is the mean of sample 2, d is the hypothesized difference between population means, and SE is the standard error.

SE = sqrt[(s12/n1) + (s22/n2)]
SE = sqrt[(15.6272/16) + (8.9212/16] = 4.5

Degree of freedom of t test is,

DF = (s12/n1 + s22/n2)2 / { [ (s12 / n1)2 / (n1 - 1) ] + [ (s22 / n2)2 / (n2 - 1) ] }
DF = (15.6272/16 + 8.9212/16)2 / { [ (15.6272 / 16)2 / (15) ] + [ (8.9212 / 16)2 / (15) ] }
DF = (20)2 / { [ (16)2 / (99) ] + [ (2)2 / (99) ] } = 400 / (2.586 + 0.162) = 23.84 = 24 (Rounding to nearest integer)

t = [ (x1 - x2) - d ] / SE = [(100.063 - 91.625) - 0] / 4.5 = 1.875

The observed difference in sample means (8.438) produced a t statistic of 1.875. We use the t Distribution Calculator to find P(t > 0.67) = 0.0365.

Interpret results. Since the P-value (0.0365) is less than the significance level (0.05), we reject the null hypothesis and conclude that the at 95% confidence level, the mean water depth before leakage of Thorndike Goggles is greater than that of OptiView Goggles.

Commentary - The mean water depth before leakage of Thorndike Goggles is greater than that of OptiView Goggles. So, the quality of Thorndike Goggles is better than that of OptiView Goggles.

Report - Based on the hypothesis test, we conclude that at 95% confidence level, the mean water depth before leakage of Thorndike Goggles is greater than that of OptiView Goggles. So, if we take multiple samples of Thorndike Goggles and OptiView Goggles, in 95% cases, the mean water depth before leakage of Thorndike Goggles is greater than that of OptiView Goggles. That is, in 95% cases, the quality of Thorndike Goggles is better than that of OptiView Goggles. If in the random sample, we find that the quality of Thorndike Goggles is not better than that of OptiView Goggles, then that may be due to sampling error or by chance. Hence, Thorndike Goggles is recommended over OptiView Goggles.

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