6.2 The central limit theorem for sums -- rrc math 1020  (Page 2/6)

References

Farago, Peter. “The Truth About Cats and Dogs: Smartphone vs Tablet Usage Differences.” The Flurry Blog, 2013. Posted October 29, 2012. Available online at http://blog.flurry.com (accessed May 17, 2013).

Chapter review

The central limit theorem tells us that for a population with any distribution, the distribution of the sums for the sample means approaches a normal distribution as the sample size increases. In other words, if the sample size is large enough, the distribution of the sums can be approximated by a normal distribution even if the original population is not normally distributed. Additionally, if the original population has a mean of μ _X and a standard deviation of σ _x , the mean of the sums is nμ _x and the standard deviation is $(\sqrt{n})$ ( σ _x ) where n is the sample size.

Formula review

The Central Limit Theorem for Sums: ∑X ~ N [( n )( μ _x ),( $\sqrt{n}$ )( σ _x )]

Mean for Sums ( ∑X ): ( n )( μ _x )

The Central Limit Theorem for Sums z -score and standard deviation for sums: $z\text{for the sample mean = }\frac{\Sigma x–(n)({\mu }_X)}{(\sqrt{n})({\sigma }_X)}$

Standard deviation for Sums ( ∑X ): $(\sqrt{n})$ ( σ _x )

Use the following information to answer the next four exercises: An unknown distribution has a mean of 80 and a standard deviation of 12. A sample size of 95 is drawn randomly from the population.

Use the following information to answer the next five exercises: The distribution of results from a cholesterol test has a mean of 180 and a standard deviation of 20. A sample size of 40 is drawn randomly.

Use the following information to answer the next six exercises: A researcher measures the amount of sugar in several cans of the same soda. The mean is 39.01 with a standard deviation of 0.5. The researcher randomly selects a sample of 100.

Use the following information to answer the next four exercise: An unknown distribution has a mean 12 and a standard deviation of one. A sample size of 25 is taken. Let X = the object of interest.

Use the following information to answer the next three exercises. A market researcher analyzes how many electronics devices customers buy in a single purchase. The distribution has a mean of three with a standard deviation of 0.7. She samples 400 customers.

Use the following information to answer the next three exercises: An unkwon distribution has a mean of 100, a standard deviation of 100, and a sample size of 100. Let X = one object of interest.