Question

16) The number of calories in a 12-ounce serving of randomly-selected regular and lite beers is listed. Is there sufficient evidence to conclude that the mean number of calories for lite beers is significantly less than that for the regular beers? Use α= 0.10. Assume the variables are approximately normally distributed. Regular: 117 118 153 107 102 152 129 126 145 138 153 124 131 28 97 103 61 109 143 116 117 96 97 Lite: a.State the hypotheses. b. Find the critical value(s). c. Compute the test value. d. Make the decision. e. Summarize the results.

Answer 1

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a) Let μί andAG be the population means and let and x2 be the means of the respective samples The summary statistics Samplel: n 13, xi- 130.4, s 17.1 Sam ple 2:%,-10, x-967, s,-31.8 The level of significance is α-010 Step1: State the null and alternative hypotheses The null and alternative hypotheses are, respectively We need to use left-tailed test Step 2: Select the distributi on to use Here, the two samples are independent, G and T2 are unknown but equal, and the sample sizes are sm all but both populations are normally distributed. Hence, all conditions are sati sfied Consequently, we will use the t - distribution b) Step3: Determine the rejection and non-rejection regions The < sign in the alternative hypothesis indicates that the test is left-tailed The significance level is 0.10. Hence, the area in left-tail distribution is α=0.10 The degrees of freedom is n+n2-2- 13+10-2 - 23-2 The critical values of t for df 21 and 0.10 area in left tail of the t - di stribution curve are -1.323188

Step 4: Calculate the value of the t- test stati stic The value of the test stati stic t for x-x2 is computed as follows 13-0)29242 (31+(10-1)1012.678 13+10-2 (12)2924231+9 (1012.678) 23-2 3509.077+9114.1 21 12623.18 21 601.1037 24.51742 5,-24.51742 24.51742.[+ 13 10 24.51742V0.076923+01 24.51742V0.176923 24.51742(0.420622) -10.31257 S10.31257 (c) Step5: The form ula for calculating the t- stati stic is (130.4-96.7)-(0 10.31257 33.7 10.31257 3.267856 3.2679 (d) Step6: Conclusion Since, the test statisti c is H 3.2679>to.D.a1-1.323188. We reject the null hypothesis Ho. Consequently, we conclude that the populaion mean 1 is less than that of the population mean2. (e) The P-value approach: P(t <t) P(e <3.2679) 0.001838 P-value 10.002 < α= 0.10 we reject the null hypothesis H . There is sufficient evidence to conclude that the mean number of calories for lite beers is significantly less than that for the regular beers at the 10% level of significance