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Chapter 10: Problem 42

What is the shape of the sampling distribution of \(\hat{p}_{1}-\hat{p}_{2}\) for two large samples? What are the mean and standard deviation of this sampling distribution?

Short Answer

Expert verified
The sampling distribution of \(\hat{p}_{1}-\hat{p}_{2}\) is normally distributed. The mean of the distribution is \(p_{1} - p_{2}\), and the standard deviation is calculated by the formula \(\sqrt{\frac{{p_{1}(1-p_{1})}}{{n_{1}}}} + \frac{{p_{2}(1-p_{2})}}{{n_{2}}}\).

Step by step solution

01

Identifying the Shape

The shape of the sampling distribution for two large samples, assuming a difference in sample proportions \(\hat{p}_{1}-\hat{p}_{2}\), follows a normal distribution, according to the Central Limit Theorem.
02

Calculating Mean of Sampling Distribution

This mean (also called the expected value, termed \(E\)) of this distribution is the difference between the two original population proportions (p1 - p2). Represented mathematically: \[E(\hat{p}_{1}-\hat{p}_{2}) = p_{1} - p_{2}\]
03

Calculating Standard Deviation of Sampling Distribution

The standard deviation of the sampling distribution (often denoted by \( \sigma \)) can be found using a standard formula that involves the sample proportions and the size of the sample. The formula being: \[ \sigma = \sqrt{\frac{{p_{1}(1-p_{1})}}{{n_{1}}}} + \frac{{p_{2}(1-p_{2})}}{{n_{2}}}\]. Here, \( n_{1} \) and \( n_{2} \) refer to the size of the first and second samples respectively, while \( p_{1} \) and \( p_{2} \) refer to the proportions of the first and second samples respectively.

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Most popular questions from this chapter

Refer to the previous exercise. Suppose Gamma Corporation decides to test governors on seven cars. However, the management is afraid that the speed limit imposed by the governors will reduce the number of contacts the salespersons can make each day. Thus, both the fuel consumption and the number of contacts made are recorded for each car/salesperson for each week of the testing period, both before and after the installation of governors. Suppose that as a statistical analyst with the company, you are directed to prepare a brief report that includes statistical analysis andinterpretation of the data. Management will use your report to help decide whether or not to install governors on all salespersons' cars. Use \(90 \%\) confidence intervals and \(.05\) significance levels for any hypothesis tests to make suggestions. Assume that the differences in fuel consumption and the differences in the number of contacts are both normally distributed.

The following information is obtained from two independent samples selected from two populations. $$ \begin{array}{lll} n_{1}=650 & \bar{x}_{1}=1.05 & \sigma_{1}=5.22 \\ n_{2}=675 & \bar{x}_{2}=1.54 & \sigma_{2}=6.80 \end{array} $$ Test at a \(5 \%\) significance level if \(\mu_{1}\) is less than \(\mu_{2}\).

The following information was obtained from two independent samples selected from two normally distributed populations with unequal and unknown population standard deviations. $$ \begin{array}{lll} n_{1}=14 & \bar{x}_{1}=.109 .43 & s_{1}=2.26 \\ n_{2}=15 & \bar{x}_{2}=.113 .88 & s_{2}=5.84 \end{array} $$ Test at a \(1 \%\) significance level if \(\mu_{1}\) is less than \(\mu_{2}\).

The following information is obtained from two independent samples selected from two normally distributed populations. $$ \begin{array}{lll} n_{1}=18 & \bar{x}_{1}=7.82 & \sigma_{1}=2.35 \\ n_{2}=15 & \bar{x}_{2}=5.99 & \sigma_{2}=3.17 \end{array} $$ a. What is the point estimate of \(\mu_{1}-\mu_{2} ?\) b. Construct a \(99 \%\) confidence interval for \(\mu_{1}-\mu_{2}\). Find the margin of error for this estimate.

According to a Bureau of Labor Statistics report released on March 25, 2015 , statisticians earn an average of \(\$ 84,010\) a year and accountants and auditors earn an average of \(\$ 73,670\) a year (www.bls. gov). Suppose that these estimates are based on random samples of 2000 statisticians and 1800 accountants and auditors. Further assume that the sample standard deviations of the annual earnings of these two groups are \(\$ 15,200\) and \(\$ 14,500\), respectively, and the population standard deviations are unknown and unequal for the two groups. a. Construct a \(98 \%\) confidence interval for the difference in the mean annual earnings of the two groups, statisticians and accountants and auditors. b. Using a \(1 \%\) significance level, can you conclude that the average annual earnings of statisticians is higher than that of accountants and auditors?
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