Two machines are used to fill plastic bottles with dishwashing detergent. The standard deviations of fill volume are known to be ?1 = 0.10 fluid ounces and ?2 = 0.15 fluid ounces for the two machines, respectively. Two random samples of n1 = 12 bottles from machine 1 and n2 = 10 bottles from machine 2 are selected, and the sample mean fill volumes are X?1 = 30.87 fluid ounces and X?2 = 30.68 fluid ounces. Assume that we have independent random samples from a normal distribution. (a) Construct a 90% two-sided confidence interval on the mean difference in fill volume. (Give your answer to 4 decimal places) ______ < ?1 ? ?2 < ______ (b) Construct a 95% two-sided confidence interval on the mean difference in fill volume. (Give your answer to 4 decimal places) ______ < ?1 ? ?2 < ______
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The formula for the standard error of the difference in means is: $$SE(\bar{X}_1 - \bar{X}_2) = \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}$$ Plugging in the given values, we get: $$SE(\bar{X}_1 - \bar{X}_2) = \sqrt{\frac{0.10^2}{12} + Show more…
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Two machines are used to fill plastic bottles with dishwashing detergent. The standard deviations of fill volume are known to be $\sigma_{1}=0.10$ fluid ounces and $\sigma_{2}=0.15$ fluid ounces for the two machines, respectively. Two random samples of $n_{1}=12$ bottles from machine 1 and $n_{2}=10$ bottles from machine 2 are selected, and the sample mean fill volumes are $\bar{x}_{1}=30.87$ fluid ounces and $\bar{x}_{2}=30.68$ fluid ounces. Assume normality. (a) Construct a $90 \%$ two-sided confidence interval on the mean difference in fill volume. Interpret this interval. (b) Construct a $95 \%$ two-sided confidence interval on the mean difference in fill volume. Compare and comment on the width of this interval to the width of the interval in part (a). (c) Construct a $95 \%$ upper-confidence interval on the mean difference in fill volume. Interpret this interval. (d) Test the hypothesis that both machines fill to the same mean volume. Use $\alpha=0.05 .$ What is the $P$ -value? (e) If the $\beta$ -error of the test when the true difference in fill volume is 0.2 fluid ounces should not exceed $0.1,$ what sample sizes must be used? Use $\alpha=0.05 .$
Statistical Inference for Two Samples
Summary Table and Roadmap for Inference Procedures for Two Samples
Two machines are used for filling plastic bottles with a net volume of 16.0 ounces. The fill volume can be assumed normal. A random sample of 10 bottles is taken from the output of each machine, and the fill volume readings of the bottles are as follows: Machine 1: 16.03, 16.01, 16.04, 13.96, 16.05, 15.98, 16.05, 16.02, 16.02, 15.99 Machine 2: 16.02, 16.03, 15.97, 16.04, 15.96, 16.02, 16.01, 16.01, 15.99, 16.00 What is the standard deviation of the sampling distribution of the sample mean for Machine 1? Find a 95% confidence interval on the difference in means of the fill volume for the two machines. Find a 90% confidence interval on the variance of the fill volume for Machine 2.
Jerelyn N.
Madhur L.
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