Additional Problems solutions

5 pz2 1 pz2 1 09772 00228 5 did you need to use the

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P(X-bar>51) = P(Z > (51-50)/0.5) = P(Z>2) = 1-P(Z<2) = 1- 0.9772 = 0.0228 5. Did you need to use the Central Limit theorem to answer #4? Why or why not? No, because the population is normal. For a normal population, the sampling distribution of X-bar is normal for all values of n. The annual rainfall in Cleveland, Ohio has an unknown distribution with mean 40.2 inches and standard deviation 8.4 inches
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6. What is the probability that next year's rainfall will be less than 40 inches? Can’t get the probability from the problem information because the population distribution is unknown here. 7. What is the probability that the average yearly rainfall over 30 years (selected at random) will be less than 40 inches? By CLT, since n=30 big enough, the sampling distribution of X-bar would be approximately normal with mean = 40.2 and std= 8.4/sqrt(30)= 1.53 P(X-bar<40 )= P(Z<(40-40.2)/1.53) = P(Z < -0.13) = 0.4483 8. Did you need to use the Central Limit Theorem for #7? Why or why not? Yes. Because the population distribution is unknown here, so we need CLT to get the sampling distribution of X-bar 9. Looking at the formula for a 95% confidence interval for the mean, suppose σ x increases (but n stays the same). What happens to each of the following parts of the confidence interval? (1 pt each; circle your answer) a. Margin of error INCREASES DECREASES STAYS SAME b. Standard deviation (aka standard error) of X-bar INCREASES DECREASES STAYS SAME c. The confidence level INCREASES DECREASES STAYS SAME d. The value of X-bar INCREASES DECREASES STAYS SAME 10. True or False 1 pt each: a. The sampling distribution of X bar always has a smaller variance than the distribution of X as long as n is greater than 1. TRUE FALSE b. The central limit theorem is always needed when you want to find the probability for X-bar TRUE FALSE c. If you want a margin of error to be cut in half, you will need a smaller sample size. TRUE FALSE n=(Z* σ /m) 2 margin error m cut in half, we would need a bigger n d. A larger |Z| value gives you a higher confidence level for a confidence interval
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TRUE FALSE
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