Question

8. For the first part of this problem, formulate the null and alternative hypotheses, carry out the appropriate test, and state the conclusion in simple terms. (a) A soda machine has been set to fill cups with 500 mL of soda each. If a random sample of 100 cups of soda from the machine yielded a mean of 502.5 mL per cup with a standard deviation of 9.5 mL, then is this sufficient evidence, at the 5 percent level of significance, to warrant resetting the machine? (b) Using the statistics in Part (a), construct a 95 percent confidence interval for u. Briefly explain why this interval should support the conclusion in Part (a).
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Answer 1

8.

a.

(1) Null and Alternative Hypotheses The following null and alternative hypotheses need to be tested: Ho: u = 500 Hapi # 500 This corresponds to a two-tailed test, for which a z-test for one mean, with known population standard deviation will be used. (2) Rejection Region Based on the information provided, the significance level is a = 0.05, and the critical value for a two-tailed test is ze = 1.96. The rejection region for this two-tailed test is R = {z: 21 > 1.96} (3) Test Statistics The Z-statistic is computed as follows: Z=- X - Mo olv 502.5 - 500 9.5/100 = 2.632 (4) Decision about the null hypothesis Since it is observed that 2 = 2.632 > Zc = 1.96, it is then concluded that the null hypothesis is rejected. Using the P-value approach: The p-value is p=0.0085, and since p= 0.0085 < 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 u is different than 500, at the 0.05 significance level.

b.

Based on the provided information, the critical z-value for a = 0.05 is ze = 1.96. The 95% confidence for the population mean u is computed using the following expression CI= (* - * 0,8 + Foto) Therefore, based on the information provided, the 95 % confidence for the population mean is CI = (502.5 – 1.50 1700-9, 502.5+ --V100 ) 1.96 x 9.5 = (502.5 – 1.862, 502.5 +1.862) = (500.638, 504.362)

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