Confidence Intervals A sample of size n = 50 is drawn from a population whose standard deviation is σ = 26. a. Find the margin of error for a 90% confidence interval for µ. M = 6.049 b. If the sample size were n = 40, would the margin of error be larger or smaller? Larger. Smaller samples (all other things remaining the same) produce larger margins of error. A sample size of n = 32 is drawn from a population whose standard deviation is σ = 12.1 a. Find the margin of error for 99% confidence interval for µ. M = 5.510 b. If the confidence level were 90%, would the margin of error be larger or smaller? Smaller. Being willing to accept a lower confidence level means we can “pin down” the mean more closely – M is smaller – the interval is narrower. A sample of size n = 10 is drawn from a normal population whose standard deviation is σ = 2.5. The sample mean is = 7.92. a. Construct a 95% confidence interval for µ. 7.92 ± 1.55 or (6.37, 9.47) We estimate with 95% confidence that the population mean is between 6.37 and 9.47. b. If the population were not approximately normal, would the confidence interval constructed in part (a) be valid? Explain. No, it would not. We can only have a valid CI for a non-Normal population when the sample size n is at least 30. A sample of size n = 80 is drawn from a normal population whose standard deviation is σ = 6.8. The sample mean is = 40.41. a. Construct a 90% confidence interval for µ. 40.41 ± 1.25 or (39.16. 41.66) We estimate with 90% confidence that the population mean is between 39.16 and 41.66. b. If the population were not approximately normal, would the confidence interval constructed in part (a) be valid? Explain. Yes; the sample size is more than 30, making the distribution of approximately Normal. The CI is approximate. A population has standard deviation σ = 17.3. a. How large a sample must be drawn so that a 95% confidence interval for µ will have a margin of error that is no more than 1.4? n = 587 b. If the required confidence level were 98%, would the necessary sample size be larger or smaller? Larger. z* would be larger, making the required n larger as well. A population has standard deviation σ = 9.2. a. How large a sample must be drawn so that a 92% confidence interval for µ will have a margin of error that is no more than 0.8? n = 406 b. If the required margin of error were 1.4, would the necessary sample size be larger or smaller? Smaller. If we are willing to accept a larger margin of error, we can get by with a smaller sample. In a simple random sample of 150 households, the sample mean number of personal computers was 1.32. Assume the population standard deviation is σ = 0.41. Construct a 95% confidence interval for the mean number of personal computers. 1.32 ± 0.07 or (1.25, 1.39) We estimate with 95% confidence that the mean number of personal computers per household is between 1.25 and 1.39. Efficiency experts study the processes used to manufacture items in order to make them as efficient as possible. One of the steps used to manufacture a metal clamp involves the drilling of three holes. In a sample of 75 clamps, the mean time to complete this step was 50.1 seconds. Assume that the population standard deviation is σ = 10 seconds. Construct a 95% confidence interval for the mean time needed to complete this step. 50.1 ± 2.3 seconds or (47.8, 52.4) seconds The estimated mean time needed is 50.1 seconds, with a 95% margin of error of ± 2.3 seconds Joe and Sally are going to construct confidence intervals from the same simple random sample. Joe’s confidence interval will have level 90% and Sally’s will have level 95%. a. Which confidence interval will have the larger margin of error? Or will they both be the same? The 95% interval will have the larger margin of error (more confidence  larger M). b. Which confidence interval is more likely to cover the population mean? Or are they both equally likely to do so? Sally’s (the 95% CI). This is because the 95% CI is constructed using a method that is accurate in about 95% of all cases. The 90% CI method has a lower “success” rate. A dean at a certain college looked up the GPA for a random sample of 85 students. The sample mean GPA was 2.82, and a 95% confidence interval for the mean GPA of all students in the college was 2.76 ˂ µ ˂ 2.88. True or false, and explain: a. We are 95% confident that the mean GPA of all students in the college is between 2.76 and 2.88. TRUE. We are using a method that is accurate about 95% of the time (in repeated sampling) so we are 95% confident in the outcome. b. We are 95% confident that the mean GPA of all students in the sample is between 2.76 and 2.88. FALSE. The mean GPA of the students in the sample is ALWAYS in the CI – it gives us the CENTER of the interval! c. The probability is 0.95 that the mean GPA of all students in the college is between 2.76 and 2.88. FALSE. This is an incorrect interpretation of what the CI means. Suppose the true mean GPA is 2.6. That is NOT in the interval, is it? So how could I say there is a probability of 0.95 that 2.6 is between 2.76 and 2.88? I can’t! d. 95% of the students in the sample had a GPA between 2.76 and 2.88. FALSE. GPAs for the students in the sample could be all over the place! The individual GPAs do not have to fall in the interval! Significance Tests A test is made of Ho: µ = 50 versus Ha: µ ˃ 50. A sample of size n = 75 is drawn, and standard deviation is σ = 20. a. Compute the value of the test statistic z. b. Is H0 rejected at the α = 0.05 level? c. Is H0 rejected at the α = 0.01 level? = 56. The population z = 2.60, so the p-value is 0.0047 Yes; p < α = 0.05 Yes; p < α = 0.01 The p-value is the probability, assuming H0 is true, of observing a value for the test statistic that is as extreme as or more extreme than the value actually observed. The smaller the P-value is, the stronger the evidence against the NULL hypothesis becomes. A test is made of Ho: µ = 14 versus Ha: µ ≠ 14. A sample of size n = 48 is drawn, and standard deviation is σ = 6. = 12. The population a. Compute the value of the test statistic z and give the p-value of the test. Z = -2.31 and p-value = 0.0208. b. Is H0 rejected at the α = 0.05 level? Yes; the p-value is < 0.05 c. Is H0 rejected at the α = 0.01 level? No; the p-value is > 0.01. A test of the hypothesis Ho: µ = 150 versus Ha: µ ˂ 150 was performed. The P-value was 0.28. Fill in the blank: If µ = 150, then the probability of observing a test statistic as extreme as or more extreme than the one actually observed is 28%. A random sample of 60 second-graders in a certain school district are given a standardized mathematics skills test. The sample mean score is = 52. Assume the standard deviation of test scores is σ = 15. The nationwide average score on this test is 50. The school superintendent wants to know whether the secondgraders in her school district have an average score that is significantly greater than the nationwide average. Use the α = 0.01 level of significance. a. State the appropriate null and alternate hypotheses. Ho: µ = 50 Ha: µ > 50 b. Compute the value of the test statistic. Z = 1.03 c. Give the p-value of the test. P-value = 0.1515 d. What is your decision about Ho? I am not going to reject Ho. This p-value is not low. e. State a conclusion. I am not convinced that the second graders in this district have a mean score that is significantly greater than the national average of 50. One of the measurements used to determine the health of a person’s lungs is the amount of air a person can exhale under force in one second. This is called the forced expiratory volume in one second, and is abbreviated FEV1. Assume the mean FEV1 for 10-year-old boys is 2.1 liters and that the population standard deviation is σ = 0.3. A random sample of 100 10-year-old boys who live in a community with high levels of ozone pollution are found to have a sample mean FEV1 of 1.95 liters. Can you conclude that the mean FEV1 in the high-pollution community is less than 2.1 liters? Use the α = 0.05 level of significance. a. State the appropriate null and alternate hypotheses. Ho: µ = 2.1 liters Ha: µ < 2.1 liters b. Compute the value of the test statistic. Z = – 5 !! Yes, that is correct c. Give the p-value of the test. P-value < 0.0002 d. What is your decision about Ho? REJECT IT!!! e. State a conclusion. There is very compelling evidence to suggest that the mean FEV1 in the highpollution community is less than 2.1 liters. A math teacher has developed a new program to help high school students prepare for the math SAT. A sample of 100 students enroll in the program. They take a math SAT exam before the program starts and again at the end to measure their improvement. The mean number of points improved was = 2.5. Assume the standard deviation is σ = 10. Let µ be the population mean number of points improved. To determine whether the program is effective, a test is made of the hypotheses H0: µ = 0 versus H1: µ ˃ 0. a. b. c. d. Compute the value of the test statistic. Z = 2.5 Compute the P-value. p-value = 0.0062 Do you reject H0 at the α = 0.05 level? Yes What is your conclusion? The mean improvement in SATM scores is significantly greater than zero. e. Is the result of practical significance? Explain. An average increase of only 2.5 points is not very practical – and may not be worth all the extra effort it would take to go through the program. The moral of the story is: Just because results are “significant” does not always mean that they are of any practical significance.