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Chapter 6: Problem 3

In general, what do the symbols \(\mu_{\bar{x}}\) and \(\sigma_{\bar{x}}\) represent? What are the values of \(\mu_{\bar{x}}\) and \(\sigma_{\bar{x}}\) for samples of size 64 randomly selected from the population of IQ scores with population mean of 100 and standard deviation of 15 ?

Short Answer

Expert verified
\(\mu_{\bar{x}}\) represents the mean of the sampling distribution: 100. \(\sigma_{\bar{x}}\) represents the standard deviation of the sampling distribution: 1.875.

Step by step solution

01

Identify the symbols

The symbols \(\mu_{\bar{x}}\) and \(\sigma_{\bar{x}}\) represent the mean and standard deviation of the sampling distribution of the sample mean, respectively.
02

Infer values for the population

We are given that the population mean \(\mu\) is 100 and the population standard deviation \(\sigma\) is 15.
03

Compute \(\mu_{\bar{x}}\)

The mean of the sampling distribution \(\mu_{\bar{x}}\) is equal to the mean of the population. Therefore, \(\mu_{\bar{x}} = \mu = 100\).
04

Compute \(\sigma_{\bar{x}}\)

The standard deviation of the sampling distribution \(\sigma_{\bar{x}}\) is given by \(\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}\), where \(\sigma\) is the population standard deviation and \ is the sample size. Substituting the given values, \(\sigma_{\bar{x}} = \frac{15}{\sqrt{64}} = \frac{15}{8} = 1.875\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

mean of sampling distribution
The mean of the sampling distribution, denoted as \(\mu_{\bar{x}}\), is an important concept in statistics. It represents the average value of the sample means over all possible samples of a specific size from a population. Interestingly, the mean of the sampling distribution is exactly the same as the population mean \(\mu\). This ensures that on average, your sample mean gives an accurate reflection of the population mean. In our exercise, since the population mean \(\mu\) is 100, the mean of the sampling distribution \(\mu_{\bar{x}}\) is also 100.
standard deviation of sampling distribution
The standard deviation of the sampling distribution, symbolized as \(\sigma_{\bar{x}}\), is a measure of how much the sample means vary from the actual population mean. To find \(\sigma_{\bar{x}}\), we calculate it using the formula \(\sigma_{\bar{x}} = \(\frac{\sigma}{\sqrt{n}}\)\) where \(\sigma\) is the population standard deviation and \(\sqrt{n}\) is the square root of the sample size. For our specific problem, with a population standard deviation of 15 and sample size of 64, the calculation is \[\sigma_{\bar{x}} = \(\frac{15}{8}\) = 1.875\]. This smaller standard deviation means that our sample means are less spread out and closer to the population mean.
Central Limit Theorem
The Central Limit Theorem (CLT) is a fundamental principle in statistics. It states that regardless of the population distribution's shape, the distribution of the sample means will be approximately normally distributed if the sample size is sufficiently large (typically \(\geq 30\)). This is very powerful because it allows us to use normal distribution principles to make inferences about sample means. In our example, we have a large sample size of 64, which fits the conditions of the CLT, ensuring that the sampling distribution of the mean will be normal.
population mean
The population mean, represented by the symbol \(\mu\), is the average of all measurements in a population. It's a central point around which the data is distributed. In real-world scenarios, knowing the population mean helps us make predictions and understand the overall trend of the data. In our problem, the population mean is provided as 100. This is crucial in determining the mean of our sampling distribution, as they share the same value.
sample size effect
Sample size plays a pivotal role in the accuracy of statistical estimates. A larger sample size generally leads to more precise estimates of the population parameters. This is because larger samples tend to average out anomalies and provide a clearer picture of the population as a whole. In our exercise, the sample size is 64, which is relatively large and therefore leads to a smaller standard deviation of the sampling distribution. Specifically, as sample size increases, the standard deviation of the sampling distribution \(\sigma_{\bar{x}}\) decreases, making our sample mean a more accurate estimate of the population mean.

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Most popular questions from this chapter

In the year that this exercise was written, there were 879 challenges made to referee calls in professional tennis singles play. Among those challenges, 231 challenges were upheld with the call overturned. Assume that in general, \(25 \%\) of the challenges are successfully upheld with the call overturned. a. If the \(25 \%\) rate is correct, find the probability that among the 879 challenges, the number of overturned calls is exactly 231 . b. If the \(25 \%\) rate is correct, find the probability that among the 879 challenges, the number of overturned calls is 231 or more. If the \(25 \%\) rate is correct, is 231 overturned calls among 879 challenges a result that is significantly high?

The U.S. Air Force once used ACES-II ejection seats designed for men weighing between \(140 \mathrm{lb}\) and \(211 \mathrm{lb}\). Given that women's weights are normally distributed with a mean of \(171.1 \mathrm{lb}\) and a standard deviation of \(46.1 \mathrm{lb}\) (based on data from the National Health Survey), what percentage of women have weights that are within those limits? Were many women excluded with those past specifications?

In testing the assumption that the probability of a baby boy is \(0.512\), a geneticist obtains a random sample of 1000 births and finds that 502 of them are boys. Using the continuity correction, describe the area under the graph of a normal distribution corresponding to the following. (For example, the area corresponding to "the probability of at least 502 boys" is this: the area to the right of \(501.5 .\) ) a. The probability of 502 or fewer boys b. The probability of exactly 502 boys c. The probability of more than 502 boys

Use the data in the table below for sitting adult males and females (based on anthropometric survey data from Gordon, Churchill, et al.). These data are used often in the design of different seats, including aircraft seats, train seats, theater seats, and classroom seats. (Hint: Draw a graph in each case.) $$ \begin{aligned} &\text { Sitting Back-to-Knee Length (inches) }\\\ &\begin{array}{l|c|c|c} \hline & \text { Mean } & \text { St. Dev. } & \text { Distribution } \\ \hline \text { Males } & 23.5 \text { in. } & 1.1 \text { in. } & \text { Normal } \\ \hline \text { Females } & 22.7 \text { in. } & 1.0 \text { in. } & \text { Normal } \\ \hline \end{array} \end{aligned} $$ Find the probability that a male has a back-to-knee length less than 21 in.

Use these parameters (based on Data Set 1 "Body Data" in Appendix B): \- Men's heights are normally distributed with mean \(68.6 \mathrm{in.}\) and standard deviation \(2.8 \mathrm{in} .\) \- Women's heights are normally distributed with mean 63.7 in. and standard deviation \(2.9 \mathrm{in}\). The Gulfstream 100 is an executive jet that seats six, and it has a doorway height of \(51.6 \mathrm{in}\). a. What percentage of adult men can fit through the door without bending? b. Does the door design with a height of \(51.6\) in. appear to be adequate? Why didn't the engineers design a larger door? c. What doorway height would allow \(40 \%\) of men to fit without bending?
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