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chapter8

# chapter8 - Chapter 8 Sampling Distributions 8.1...

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412 Chapter 8 Sampling Distributions 8.1 Distribution of the Sample Mean 1. The sampling distribution of a statistic (such as the sample mean) is the probability distribution for all possible values of the statistic computed from samples of fixed size, n . 2. The Central Limit Theorem states that the sampling distribution of the sample mean, x , becomes increasingly more normal as the sample size n increases (provided X σ is finite). 3. standard error of the mean 4. zero 5. The mean of the sampling distribution of x is given by x μ = and the standard deviation is given by x n = . 6. To say that the sampling distribution of x is normal, we would require that the population be normal. That is, the distribution of X must be normal. 7. four; To see this, note that 1 2 4 x nn == . 8. True; assuming we are drawing random samples, these two results are true regardless of the distribution of the population. 9. The sampling distribution would be exactly normal. The mean and standard deviation would be 30 x and 8 2.53 10 x n . 10. No, because the sample size is large, i.e. 40 30 n = . If the distribution of the population is not normal, the sampling distribution is approximately normal with mean 50 x and 42 0.63 40 10 x n = = . Note: The answers to the exercises in this section are based on values from the standard normal table. Since answers computed using technology do not involve rounding of Z -scores, they will often differ from the answers given below in the third and fourth decimal places.

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Section 8.1 Distribution of the Sample Mean 413 11. 80 x μ == ; 14 14 2 7 49 x n σ = = 12. 64 x ; 18 18 3 6 36 x n = = 13. 52 x ; 10 2.182 21 x n 14. 27 x ; 6 1.549 15 x n 15. (a) Since 80 = and 14 = , the mean and standard deviation of the sampling distribution of x are given by: 80 x ; 14 14 2 7 49 x n = = We are not told that the population is normally distributed, but we do have a large sample size ( 30 n ). Therefore, we can use the Central Limit Theorem to say that the sampling distribution of x is approximately normal. (b) () 83 80 83 1.50 1 1.50 1 0.9332 0.0668 2 Px PZ ⎛⎞ >= > = > = = = ⎜⎟ ⎝⎠ (c) 75.8 80 75.8 2.10 0.0179 2 ≤=≤ = = (d) 78.3 80 85.1 80 78.3 85.1 0.85 2.55 22 0.9946 0.1977 0.7969 P Z P Z −− << = = − =−= 16. (a) Since 64 = and 18 = , the mean and standard deviation of the sampling distribution of x are given by: 64 x ; 18 18 3 6 36 x n = = We are not told that the population is normally distributed, but we do have a large sample size ( 30 n ). Therefore, we can use the Central Limit Theorem to say that the sampling distribution of x is approximately normal. (b) 62.6 64 62.6 0.47 0.3192 3 <=< = < = (c) 68.7 64 68.7 1.57 1 1.57 3 1 0.9418 0.0582 ≥= = < =− =
Chapter 8 Sampling Distributions 414 (d) () 59.8 64 65.9 64 59.8 65.9 1.40 0.63 33 0.7357 0.0808 0.6549 Px P x P Z −− ⎛⎞ << = = − ⎜⎟ ⎝⎠ =−= 17. (a) The population must be normally distributed. If this is the case, then the sampling distribution of x is exactly normal. The mean and standard deviation of the sampling distribution are 64 x μ == and 17 4.907 12 x n σ .

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chapter8 - Chapter 8 Sampling Distributions 8.1...

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