Instructions: Complete The Following Exercises Located At Th

Instructions: Complete the following exercises located at the end of each chapter and put them into a Word document to be submitted as directed by the instructor.

Complete the following exercises located at the end of each chapter and put them into a Word document to be submitted as directed by the instructor. Show all relevant work; use the equation editor in Microsoft Word when necessary. Chapter 9, numbers 9.7, 9.8, 9.9, 9.13, and 9.14 9.7 Define the sampling distribution of the mean. 9.8 Specify three important properties of the sampling distribution of the mean. 9.9 Indicate whether the following statements are true or false. If we took a random sample of 35 subjects from some population, the associated sampling distribution of the mean would have the following properties: (a) Shape would approximate a normal curve. (b) Mean would equal the one sample mean. (c) Shape would approximate the shape of the population. (d) Compared to the population variability, the variability would be reduced by a factor equal to the square root of 35. (e) Mean would equal the population mean. (f) Variability would equal the population variability. 9.13 Given a sample size of 36, how large does the population standard deviation have to be for the standard error to be (a) 1? (b) 2? (c) 5? (d) 100? 9.14 (a) A random sample of size 144 is taken from the local population of grade-school children. Each child estimates the number of hours per week spent watching TV. At this point, what can be said about the sampling distribution? (b) Assume that a standard deviation, σ, of 8 hours describes the TV estimates for the local population of schoolchildren. At this point, what can be said about the sampling distribution? (c) Assume that a mean, µ, of 21 hours describes the TV estimates for the local population of schoolchildren. Now what can be said about the sampling distribution? (d) Roughly speaking, the sample means in the sampling distribution should deviate, on average, about ___ hours from the mean of the sampling distribution and from the mean of the population. (e) About 95 percent of the sample means in this sampling distribution should be between ___ hours and ___ hours.

Paper For Above instruction

The sampling distribution of the mean is a fundamental concept in inferential statistics, describing the distribution of sample means obtained from all possible samples of a specific size drawn from a population. It is essential for understanding how sample estimates relate to population parameters. According to standard statistical theory, the sampling distribution of the mean provides insights into the variability and shape of the distribution of sample means under repeated sampling.

Three critical properties characterize the sampling distribution of the mean. Firstly, its distribution tends to be approximately normal, especially as the sample size increases, regardless of the population’s distribution, a consequence of the Central Limit Theorem. Secondly, the mean of the sampling distribution (called the expected value) equals the population mean, making it an unbiased estimator. Thirdly, the standard deviation of the sampling distribution, known as the standard error, depends on the population standard deviation and the sample size, decreasing as the sample size increases, which implies that larger samples tend to produce more precise estimates of the population mean.

Regarding sample size and shape, when drawing a sample of 35 subjects from a population, certain properties of the sampling distribution can be anticipated. If the population distribution is normal, then the sampling distribution of the mean will also be normal, regardless of sample size. When the population distribution is not normal but the sample size is sufficiently large (typically n ≥ 30), the sampling distribution will still approximate normality due to the Central Limit Theorem. The mean of the sampling distribution will be equal to the population mean, and the variability will be reduced relative to the population by a factor of the square root of the sample size, illustrating the Law of Large Numbers's effect in reducing sampling variability.

To analyze the standard error, if a sample of size 36 is drawn, the population standard deviation (σ) must be large enough for the standard error (SE) to reach certain levels. The standard error is calculated by σ divided by the square root of the sample size. So, for SE = 1, σ must be 6; for SE = 2, σ must be 12; for SE=5, σ is 30; and for SE=100, σ equals 600. These calculations underscore the direct relationship between population variability, sample size, and the precision of the sampling distribution.

When examining a large sample of 144 grade-school children estimating weekly TV hours, the sampling distribution of the mean estimate can be characterized as approximately normal under most circumstances, especially considering the large sample size. Assuming the population standard deviation (σ) is 8 hours, the standard error would be 8 divided by the square root of 144, which is 8/12 = 0.6667 hours. This small standard error indicates that sample means will cluster closely around the true population mean, 21 hours, helping ensure reliable estimates. If the population mean is indeed 21 hours, then the sampling distribution will center around this value, with the variation determined by the standard error.

On average, the deviation of sample means from the population mean would be approximately equal to the standard error, which in this case is about 0.67 hours. To estimate where 95% of the sample means fall, assuming normality, we use approximately two standard errors (about 1.96) on either side of the mean. Therefore, the 95% confidence interval would be roughly 21 hours ± (1.96 × 0.67), resulting in an interval from approximately 19.7 to 22.3 hours. This interval provides a range within which we can be confident that the true population mean of TV watching hours lies.

In hypotheses testing, the null hypothesis might propose that the mean TV watching time is 21 hours. A sample mean significantly different from this value could lead to rejection of the null hypothesis at a specified significance level (e.g., 0.05), especially when the standard error is small. These principles apply similarly across other research contexts, such as testing medical hypotheses or educational policies, where sample data inform conclusions about population parameters.

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