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Chapter 10: Problem 2

Describe the sampling distribution of \(\bar{x}_{1}-\bar{x}_{2}\) for two independent samples when \(\sigma_{1}\) and \(\sigma_{2}\) are known and either both sample sizes are large or both populations are normally distributed. What are the mean and standard deviation of this sampling distribution?

Short Answer

Expert verified
The mean of the sampling distribution of \(\bar{x}_{1}-\bar{x}_{2}\) is \(\mu_{1} - \mu_{2}\). The standard deviation is calculated using this formula \(\sqrt{\frac{\sigma_{1}^{2}}{n_{1}} + \frac{\sigma_{2}^{2}}{n_{2}}}\).

Step by step solution

01

Formulating the Mean

The mean of the sampling distribution of the difference between two independent sample means, also called the expected value, can be calculated as the difference between the population means. Therefore, if \(\mu_{1}\) is the population mean for sample 1 and \(\mu_{2}\) is the population mean for sample 2, then: \[E(\bar{x}_{1} - \bar{x}_{2}) = \mu_{1} - \mu_{2}\]
02

Formulating the Standard Deviation

The standard deviation of the sampling distribution of the difference between two means, often referred to as the standard error, is given by the formula: \[\sigma(\bar{x}_{1} - \bar{x}_{2}) = \sqrt{\frac{\sigma_{1}^{2}}{n_{1}} + \frac{\sigma_{2}^{2}}{n_{2}}}\]where \(\sigma_{1}\) and \(\sigma_{2}\) are the population standard deviations for samples 1 and 2, and \(n_{1}\) and \(n_{2}\) are the sizes of the two samples respectively.

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Most popular questions from this chapter

Assuming that the two populations have unequal and unknown population standard deviations, construct a \(99 \%\) confidence interval for \(\mu_{1}-\mu_{2}\) for the following. $$ \begin{array}{lll} n_{1}=48 & \bar{x}_{1}=.863 & s_{1}=.176 \\ n_{2}=46 & \bar{x}_{2}=.796 & s_{2}=.068 \end{array} $$

When are the samples considered large enough for the sampling distribution of the difference between two sample proportions to be (approximately) normal?

A company sent seven of its employees to attend a course in building self- confidence. These employees were evaluated for their self-confidence before and after attending this course. The following table gives the scores (on a scale of 1 to 15,1 being the lowest and 15 being the highest score) of these employees before and after they attended the course. $$ \begin{array}{l|rrrrrrr} \hline \text { Before } & 8 & 5 & 4 & 9 & 6 & 9 & 5 \\ \hline \text { After } & 10 & 8 & 5 & 11 & 6 & 7 & 9 \\ \hline \end{array} $$ a. Construct a \(95 \%\) confidence interval for the mean \(\mu_{d}\) of the population paired differences, where a paired difference is equal to the score of an employee before attending the course minus the score of the same employee after attending the course b. Test at the \(1 \%\) significance level whether attending this course increases the mean score of employees.

In a Pew Research survey from July 2009, participants were asked about the economic news that they had heard since May \(2009 .\) Forty-five percent of Independents stated that the news was mostly bad, as opposed to \(48 \%\) of Republicans and \(30 \%\) of Democrats. The study mentioned a total sample size of 1001\. Assume that this sample included 395,286 , and 320 Independents, Republicans, and Democrats, respectively. a. Construct a \(95 \%\) confidence interval for the difference in the population proportions for each of the three pairs of political affiliations. b. At the \(1 \%\) level, can you conclude that the percentage of Independents who stated that the news was mostly bad was significantly different from the corresponding percentage of Republicans? Use the critical value method. c. Repeat part b using the \(p\) -value method.

The management at New Century Bank claims that the mean waiting time for all customers at its branches is less than that at the Public Bank, which is its main competitor. A business consulting firm took a sample of 200 customers from the New Century Bank and found that they waited an average of \(4.5\) minutes before being served. Another sample of 300 customers taken from the Public Bank showed that these customers waited an average of \(4.75\) minutes before being served. Assume that the standard deviations for the two populations are \(1.2\) and \(1.5\) minutes, respectively. a. Make a \(97 \%\) confidence interval for the difference between the two population means. b. Test at the \(2.5 \%\) significance level whether the claim of the management of the New Century Bank is true. c. Calculate the \(p\) -value for the test of part b. Based on this \(p\) -value, would you reject the null hypothesis if \(\alpha=.01 ?\) What if \(\alpha=.05 ?\)
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