Question

2. The resistance of coils manufactured by a certain company is claimed to have a mean value of 100 ?. A sample of 30 coils is taken an it is found that the sample mean is 115 ? and the sample standard deviation is 2012 (a) Is the claim justified if a 95% confidence level is used? (b) Is the claim justified if a 90% confidence level is used?

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Answer 1

a)

The following null and alternative hypotheses need to be tested: Ho: μ = 100 Ha: μ#100 This corresponds to a two-tailed test, for which a t-test for one mean, with unknown population standard deviation will be used. Based on the information provided, the significance level is a = 0.05, and the critical value for a two-tailed test is tc 2.045. The rejection region for this two-tailed test is R = {t: t > 2.045} The t-statistic is computed as follows: X. ,句 115 100 4108 鞋 Since it is observed that t = 4.108 > t.-2045, it is then concluded that the null hypothesis is rejected Using the P-value approach: The p-value is p = 0.0003, and since p = 0.0003 < 0.05, it is concluded that the null hypothesis is rejected. (5) Conclusion It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that the population mean μ is different than 100, at the 0.05 significance level. Confidence Interval The 95% confidence interval is 107.532 μ 122.468

b)

Nu The following null and alternative hypotheses need to be tested Ho: μ 100 Ha: μ 100 This corresponds to a two-tailed test, for which a t-test for one mean, with unknown population standard deviation will be used (2) Rejection Region Based on the information provided, the significance level is 0.1, and the critical value for a two- tailed test is t 1.699 The rejection region for this two-tailed test is R = {t : |t| > 1.699) The t-statistic is computed as follows 140 115 100 Since it is observed that t = 4.108 > te = 1.699, it is then concluded that the null hypothesis is rejected Using the P-value approach: The p-value is p 0.0003, and since p 0.0003 concluded that the null hypothesis is rejected 0.1, it is It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that the population mean μ is different than 100 at the 0.1 significance level. The 90% confidence interval is 108.796 121.204.