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Chapter 7: Problem 14

The living spaces of all homes in a city have a mean of 2300 square feet and a standard deviation of 500 square feet. Let \(\bar{x}\) be the mean living space for a random sample of 25 homes selected from this city. Find the mean and standard deviation of the sampling distribution of \(\bar{x}\).

Short Answer

Expert verified
The mean of the sampling distribution of \( \bar{x} \) is 2300 square feet, and the standard deviation is 100 square feet.

Step by step solution

01

Find Mean of Sampling Distribution

The mean of a sampling distribution, often denoted as \( \mu_{\bar{x}} \), is equal to the mean of the population. So in this case, \( \mu_{\bar{x}} \) would be equal to 2300 square feet.
02

Find Standard Deviation of Sampling Distribution

The standard deviation of a sampling distribution, also known as the standard error, is calculated by dividing the standard deviation of the population by the square root of the sample size. This can be represented as \( \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} \), where \( \sigma \) is the standard deviation of the population and \( n \) is the sample size. Plugging in the given values, we get \( \sigma_{\bar{x}} = \frac{500}{\sqrt{25}} = 100 \) square feet.

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Most popular questions from this chapter

According to a Gallup poll conducted April \(3-6,2014,21 \%\) of Americans aged 18 to 29 said that college loans and/or expenses were the top financial problem facing their families. Assume that this percentage is true for the current population of Americans aged 18 to \(29 .\) Let \(\hat{p}\) be the proportion in a random sample of 900 Americans aged 18 to 29 who hold the above opinion. Find the mean and standard deviation of the sampling distribution of \(\hat{p}\) and describe its shape.

The Toyota Prius hybrid car is estimated to get an average of 50 miles per gallon (mpg) of gas. However, the gas mileage varies from car to car due to a variety of conditions, driving styles, and other factors and has been reported to be as high as \(70 \mathrm{mpg}\). Suppose that the distribution of miles per gallon for Toyota Prius hybrid cars has a mean of \(50 \mathrm{mpg}\) and a standard deviation of \(5.9 \mathrm{mpg}\). Find the probability that the average miles per gallon for 38 randomly selected Prius hybrid cars is a. more than \(51.5\) b. between 48 and 51 c. less than 53 d. greater than the population mean by \(2.5\) or more

Johnson Electronics Corporation makes electric tubes. It is known that the standard deviation of the lives of these tubes is 150 hours. The company's research department takes a sample of 100 such tubes and finds that the mean life of these tubes is 2250 hours. What is the probability that this sample mean is within 25 hours of the mean life of all tubes produced by this company?

Explain the central limit theorem.

According to the central limit theorem, the sampling distribution of \(\hat{p}\) is approximately normal when the sample is large. What is considered a large sample in the case of the proportion? Briefly explain.
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