Rec 11 - Sampling Distributions and Confidence Intervals

STAT 1430 Recitation 11Sampling Distributions and Confidence Intervals PART 1: SAMPLING DISTRIBUTIONS 1. As n increases, which of the following statements is true? a. The standard error of the sample mean decreases. b. The population standard deviation decreases. c. The population standard deviation increases. d. The standard error of the sample mean increases. 2. The mean of X equals the mean of X when? a. Only if X has a normal distribution b. Only if n is large c. Both a and b are needed d. This statement is always true no matter what. 3. Let X be the sample mean from X ’s distribution. The standard error of X is: a. The same as the standard deviation of X b. Greater than the standard deviation of X c. Less than the standard deviation of X 4. If X has a normal distribution then X has a normal distribution no matter what n is. a. True b. False 5. Suppose test scores (X) have a normal distribution with mean 70 and standard deviation 5. a. If you sample one test score at random, what is the chance that it will be less than 65? b. If you sample 36 test scores at random, what is the chance that the average will be less than 65? Use your lecture notes: 6. How large does n have to be for the CLT to take effect, assuming X did not have a normal distribution? a. n at least 30 b. np at least 10 and n(1-p) at least 10 c. n can be any number d. None of these 7. What aspects of ¯ X does the CLT apply to? a. Shape b. Mean

STAT 1430 Recitation 11Sampling Distributions and Confidence Intervals c. Standard Error 8. The Central Limit Theorem states that, if a random sample of size n is drawn from a population, then the sampling distribution of the sample mean ¯ X : a. Is approximately normal if n >= 30 b. Is approximately normal if n < 30 c. Is approximately normal if the underlying population is normal d. Has the same variance as the population 9. If X has a normal distribution, could you find a probability for ¯ X if n was only 10? a. Yes, because X has a normal distribution already. b. No, because you always need n to be at least 30. 10. Let ¯ X be the sample mean from X ’s distribution. Suppose X does not have a normal distribution. The distribution (shape) of ¯ X is: a. Exactly normal b. Approximately normal for any sample size by the Central Limit Theorem c. Approximately normal for large enough samples by the Central Limit Theorem d. None of the above 11. The Central Limit Theorem is important in statistics because: a. For n large, it says the distribution of the sample mean is approximately normal, regardless of the shape of the population. b. For any sample size, it says the sampling distribution of the sample mean is approximately normal. c. For n large, it says the distribution of the sample mean is exactly normal, regardless of the shape of the population. d. For any sample size, it says the sampling distribution of the sample mean is exactly normal 12. Suppose X has a normal distribution and ¯ X represents the average of a sample of size n. Which of the following is true? a. ¯ X has an approximate normal distribution for any value of n b. ¯ X has an approximate normal distribution if n is large enough. c. ¯ X has an exact normal distribution for any value of n. d. ¯ X has an exact normal distribution if the sample size is large enough. 13. Suppose the age of the customers at post office has a non-normal distribution with mean 40 and standard deviation 5 years. i. Select 100 customers at random. What is the distribution of their average age? Describe its shape, its mean, and standard error.

STAT 1430 Recitation 11Sampling Distributions and Confidence Intervals 1. A confidence interval provides a range of ___________ values for a population parameter (such as the population mean.) 2. A confidence interval for the mean equals x plus or minus “something”. The “something” is called the _________________________. 3. What 3 items affect the margin of error of a confidence interval? a. b. c. 4. As n increases, what happens to the margin of error (MOE) of your confidence interval? a. Increases b. Decreases c. Stays the same Explain why this happens using plain language (do not use a math argument). 5. As the confidence level of your confidence interval increases, what happens to the value of Z*? a. Increases b. Decreases c. Stays the same Explain why this happens using plain language (do not use a math argument). 6. As the standard deviation of the population increases, what happens to the MOE? a. Increases b. Decreases c. Stays the same Explain why this happens using plain language (do not use a math argument). 7. Suppose a 95% confidence interval for a population mean was found to be (60, 80). Can we find the sample mean? Choose yes/no and justify your answer. For the next 2 problems: A group of randomly selected 50 sales representatives from a company this year had an average sales of $107.5 million this year. Assume the standard deviation of sales is stable at $20.5 million. 8. Use your data to find a range of likely values for average sales for all representative in the entire company using StatCrunch . Verify with your own calculation s and show your work. FINDING A CONFIDENCE INTERVAL FOR THE POPULATION MEAN USING STATCRUNCH. In this case you know the population standard deviation σ. --Mystatlab.com / StatCrunch / Open StatCrunch. --Go to “STATS/Z STATS/ONE SAMPLE/ WITH SUMMARY.

STAT 1430 Recitation 11Sampling Distributions and Confidence Intervals --In the window, type in the mean, standard deviation, and sample size. (Note it says sample standard deviation but it should say population standard deviation.) --Hit COMPUTE! You will get the Upper Limit and the Lower Limit of your confidence interval. Include a statement with it. 9. The average sales for the entire company last year was reported to be $100 million. According to your results from the previous problem, do you think the average sales have changed this year? Why or why not? If so, how has it changed? (Hint: Think about what your range of likely values is for average sales.) 10. Scores on an MBA placement exam are reported to have a normal distribution with standard deviation 18. The exam officials stated that the average score for all students was 70. You take a random sample of 50 students and find their average score is 67. Use your data to estimate the mean score for all students taking the MBA placement exam using StatCrunch . Verify your answer using calculations and show your work. 11. Students at the union want to estimate the average number of ounces of coffee in a cup. They take a random sample of 40 cups and find the mean is 5.2 ounces. Assume amount dispensed has a normal distribution and that the standard deviation is set at 0.24 ounces per cup. Find your best estimate for the average amount of coffee being dispensed by this machine. Use StatCrunch. Verify your answer using calculations and show your work. FOR THE NEXT 4 PROBLEMS: You want to estimate the average household income for Ohio. You want the margin of error to be no more than $1,000. Prior data shows the standard deviation of household income is $30,000. 12. How many households should we sample to achieve the desired margin of error? 13. Suppose you want to cut the margin of error down to $500 next time. What should your sample size be? 14. Suppose you want to cut the margin down to 1/3 of what it started at (1/3 of 1000). What does your sample sizes have to be? (Be careful!) 15. Why do we always round up when finding the appropriate sample size to achieve a certain margin of error, even when the value after the decimal point is less than .5? (For example, if we solve for n and get 422.2 households, why do we round this up to 423 households when reporting the required n?) FOR THE NEXT 2 PROBLEMS: Suppose you want to estimate the average number of years employees have worked at your company so far. You take a random sample of 100 workers and you find the average