7.1 The central limit theorem for sample means (averages)  (Page 3/18)

References

Baran, Daya. “20 Percent of Americans Have Never Used Email.”WebGuild, 2010. Available online at http://www.webguild.org/20080519/20-percent-of-americans-have-never-used-email (accessed May 17, 2013).

Data from The Flurry Blog, 2013. Available online at http://blog.flurry.com (accessed May 17, 2013).

Data from the United States Department of Agriculture.

Chapter review

In a population whose distribution may be known or unknown, if the size ( n ) of samples is sufficiently large, the distribution of the sample means will be approximately normal. The mean of the sample means will equal the population mean. The standard deviation of the distribution of the sample means, called the standard error of the mean, is equal to the population standard deviation divided by the square root of the sample size ( n ).

Formula review

The Central Limit Theorem for Sample Means: $\overline{X}$ ~ N $\left({\mu }_x\text{, }\frac{\sigma x}{\sqrt{n}}\right)$

The Mean $\overline{X}$ : μ _x

Central Limit Theorem for Sample Means z-score and standard error of the mean: $z=\frac{\overline{x}-{\mu }_x}{\left(\frac{{\sigma }_x}{\sqrt{n}}\right)}$

Standard Error of the Mean (Standard Deviation ( $\overline{X}$ )): $\frac{{\sigma }_x}{\sqrt{n}}$

Use the following information to answer the next six exercises: Yoonie is a personnel manager in a large corporation. Each month she must review 16 of the employees. From past experience, she has found that the reviews take her approximately four hours each to do with a population standard deviation of 1.2 hours. Let Χ be the random variable representing the time it takes her to complete one review. Assume Χ is normally distributed. Let $\overline{X}$ be the random variable representing the mean time to complete the 16 reviews. Assume that the 16 reviews represent a random set of reviews.