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Chapter 8: Problem 13

Suppose that a random sample of size 64 is to be selected from a population with mean 40 and standard deviation 5 . a. What are the mean and standard deviation of the \(\bar{x}\) sampling distribution? Describe the shape of the \(\bar{x}\) sampling distribution. b. What is the approximate probability that \(\bar{x}\) will be within \(0.5\) of the population mean \(\mu\) ? c. What is the approximate probability that \(\bar{x}\) will differ from \(\mu\) by more than \(0.7 ?\)

Short Answer

Expert verified
a. The mean of the \(\bar{x}\) sampling distribution is 40, the standard deviation of the \(\bar{x}\) sampling distribution is 0.625, and the shape is approximately normal. b. The approximate probability that \(\bar{x}\) will be within 0.5 of the population mean \(\mu\) is 0.5764. c. The approximate probability that \(\bar{x}\) will differ from \(\mu\) by more than 0.7 is 0.484.

Step by step solution

01

Mean and Standard deviation of the \(\bar{x}\)

The mean of the sampling distribution (which is the mean of the means) is equal to the population mean \(\mu\), which in this case is 40. The standard deviation of the sampling distribution, also called the standard error \(SE\), is equal to the standard deviation of the population \(\sigma\) divided by the square root of the sample size \(n\), so: \(SE = \sigma / \sqrt{n} = 5 / \sqrt{64} = 5 / 8 = 0.625.\)
02

Describe the shape of the sampling distribution

The Central Limit Theorem tells us that the sampling distribution will be approximately normal when the sample size \(n\) is large, typically \(n > 30\). In this case, our sample size is 64, which is large, so we can say the sampling distribution will be approximately normal in shape.
03

Probabilities of \(\bar{x}\) being within a certain range

To find the probabilities, first convert the range to z-scores. A z-score tells us how many standard deviations away from the mean the value is. Use the formula: \(Z = (\bar{x} - \mu) / SE \). \nFor part b, the range is from 39.5 to 40.5. So calculate the z-scores: \(Z_{40.5} = (40.5 - 40) / 0.625 = 0.8\) and \(Z_{39.5} = (39.5 - 40) / 0.625 = -0.8\). The probability that \(\bar{x}\) is between these two values is the area under the standard normal curve between these two z-scores. This value is approximately 0.5764.\nFor part c, in a similar fashion, calculate the z-scores for range \(\mu - 0.7\) and \(\mu + 0.7\). Then, find the area under the curve, which represents the probability. The area is 1 minus the combined area for those two z-scores. That is, \(1 - 2*P(Z_{0.7})\). Since \(P(Z_{0.7})\) is 0.758, the result is approximately 0.484.

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Most popular questions from this chapter

Consider the following population: \(\\{2,3,3,4,4\\}\). The value of \(\mu\) is \(3.2\), but suppose that this is not known to an investigator, who therefore wants to estimate \(\mu\) from sample data. Three possible statistics for estimating \(\mu\) are Statistic \(1:\) the sample mean, \(\bar{x}\) Statistic \(2:\) the sample median Statistic \(3 :\) the average of the largest and the smallest values in the sample A random sample of size 3 will be selected without replacement. Provided that we disregard the order in which the observations are selected, there are 10 possible samples that might result (writing 3 and \(3^{*}, 4\) and \(4^{*}\) to distinguish the two 3's and the two t's in the population): $$ \begin{array}{lllll} 2,3,3^{*} & 2,3,4 & 2,3,4^{*} & 2,3^{*}, 4 & 2,3^{*}, 4^{*} \\ 2,4,4^{*} & 3,3^{*}, 4 & 3,3^{*}, 4^{*} & 3,4,4^{*} & 3^{*}, 4,4^{*} \end{array} $$ For each of these 10 samples, compute Statistics 1,2 , and 3. Construct the sampling distribution of each of these statistics. Which statistic would you recommend for estimating \(\mu\) and why?

The article "Thrillers" (Newsweek, April 22,1985) stated, "Surveys tell us that more than half of America's college graduates are avid readers of mystery novels." Let \(p\) denote the actual proportion of college graduates who are avid readers of mystery novels. Consider a sample proportion \(\hat{p}\) that is based on a random sample of 225 college graduates. a. If \(p=.5\), what are the mean value and standard deviation of \(\hat{p} ?\) Answer this question for \(p=.6\) Does \(\hat{p}\) have approximately a normal distribution in both cases? Explain. b. Calculate \(P(\hat{p} \geq .6)\) for both \(p=.5\) and \(p=.6\). c. Without doing any calculations, how do you think the probabilities in Part (b) would change if \(n\) were 400 rather than \(225 ?\)

Explain the difference between a population characteristic and a statistic.

The thickness (in millimeters) of the coating applied to disk drives is one characteristic that determines the usefulness of the product. When no unusual circumstances are present, the thickness \((x)\) has a normal distribution with a mean of \(2 \mathrm{~mm}\) and a standard deviation of \(0.05 \mathrm{~mm}\). Suppose that the process will be monitored by selecting a random sample of 16 drives from each shift's production and determining \(\bar{x}\), the mean coating thickness for the sample. a. Describe the sampling distribution of \(\bar{x}\) (for a sample of size 16). b. When no unusual circumstances are present, we expect \(\bar{x}\) to be within \(3 \sigma_{\bar{x}}\) of \(2 \mathrm{~mm}\), the desired value. An \(\bar{x}\) value farther from 2 than \(3 \sigma_{\bar{x}}\) is interpreted as an indication of a problem that needs attention. Compute \(2 \pm 3 \sigma_{\bar{x}} .\) (A plot over time of \(\bar{x}\) values with horizontal lines drawn at the limits \(\mu \pm 3 \sigma_{x}^{-}\) is called a process control chart.) c. Referring to Part (b), what is the probability that a sample mean will be outside \(2 \pm 3 \sigma_{\bar{x}}^{-}\) just by chance (that is, when there are no unusual circumstances)? d. Suppose that a machine used to apply the coating is out of adjustment, resulting in a mean coating thickness of \(2.05 \mathrm{~mm}\). What is the probability that a problem will be detected when the next sample is taken? (Hint: This will occur if \(\bar{x}>2+3 \sigma_{\bar{x}}\) or \(\bar{x}<2-3 \sigma_{\bar{x}}\) when \(\left.\mu=2.05 .\right)\)

Explain the difference between \(\sigma\) and \(\sigma_{\bar{x}}\) and between \(\mu\) and \(\mu_{\bar{x}}^{-\text {. }}\)
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