6.1 HW

Find the critical value zc necessary to form a confidence interval at the level of confidence shown below. c=0.89 zc= _

1.60 1-0.89=0.11 0.11/2=0.055 0.055=-1.60

Find the critical value zc necessary to form a confidence interval at the level of confidence shown below. c=0.91 zc= _

1.70 1-0.91=0.09 0.09/2=0.045 0.045=-1.70

ANSWER a. You are more likely to be correct using an interval estimate because it is unlikely that a point estimate will exactly equal the population mean

QUESTION When estimating a population mean are you more likely to be correct when you use a point estimate or an interval estimate? Explain your reasoning. Choose the correct answer below. a. You are more likely to be correct using an interval estimate because it is unlikely that a point estimate will exactly equal the population mean b. If n≤30 an interval estimate is more accurate. If n>30 a point estimate is more accurate c. You are more likely to be correct using a point estimate because an is too broad and contains many possible values d. There is no difference between an interval estimate and a point estimate in terms of accuracy

ANSWER 16 (b) 26.85; 27.15; does; inside 0.6/*square root*107=0.058 0.058x2.575=0.1494 27-0.1494=26.85 27+0.1494=27.15

QUESTION 16 (b) The 99% confidence interval for a sample size of 117 is (_, _). It _ seem possible that the population mean could be less than 27.1 inches because values below 27.1 inches fall _ the confidence interval.

ANSWER 16 (a) 107 (2.575x0.6/0.15)^2=106.09=107 (b) see slide 20

QUESTION 16 A soccer ball manufacturer wants to estimate the mean circumference of soccer balls within 0.15 inch. ​(a) Determine the minimum sample size required to construct a 99​% confidence interval for the population mean. Assume the population standard deviation is 0.6 inch. ​(b) The sample mean is 27 inches. With a sample size of 117​, a 99​% level of​ confidence, and a population standard deviation of 0.6 ​inch, does it seem possible that the population mean could be less than 27.1 ​inches? Explain. (a) The minimum sample size required to construct a 99​% confidence interval is _ soccer balls (b) see slide 20

ANSWER 17 (a) will increase (b) will decrease (c) will increase

QUESTION 17 If all other quantities remain the​ same, how does the indicated change affect the minimum sample size​ requirement? ​(a) Increase in the level of confidence ​(b) Increase in the error tolerance ​(c) Increase in the population standard deviation (a) How does an increase in the level of confidence affect the minimum sample size​ requirement? Choose the correct answer below. An increase in the level of confidence _ the minimum sample size required (b) How does an increase in the error tolerance affect the minimum sample size​ requirement? Choose the correct answer below. An increase in the error tolerance _ the minimum sample size required (c) How does an increase in the population standard deviation affect the minimum sample size​ requirement? Choose the correct answer below. An increase in the population standard deviation _ the minimum sample size required

For the same sample​ statistics, which level of confidence would produce the widest confidence​ interval? Explain your reasoning. Choose the correct answer below. a. 90%, because as the level of confidence​ decreases, zc decreases b. 90%, because as the level of confidence​ decreases, zc increases c. 99%, because as the level of confidence​ increases, zc decreases ​d. 99%, because as the level of confidence​ increases, zc increases

d. 99%, because as the level of confidence​ increases, zc increases

Type in your answer. What is a single value estimate for a population parameter called? _

point estimate

Use the values on the number line to find the sampling error. The sampling error is _

-0.65 4.1-4.75=-0.65

Watch the following video and answer the questions below: https://news.yahoo.com/blogs/ticket/poll-obama-romney-tied-among-likely-voters-ohio-152641128--election.html?guccounter=1 1) What is Obama's percent? _% 2) What is Romney's percent? _% 3) The margin of error is +- _%

1) 45% 2) 44% 3) 3.5% *3% other candidates *8% undecided

Use the confidence interval to find the estimated margin of error. Then find the sample mean. A biologist reports a confidence interval of (3.7, 4.1) when estimating the mean height​ (in centimeters) of a sample of seedlings. (a) The estimated margin of error is _ (b) The sample mean is _

(a) 0.2 3.7+4.1=7.8 7.8/2=3.9 3.9-3.7=0.2 (b) 3.9

For each level of confidence c​ below, determine the corresponding normal confidence interval. Assume each confidence interval is constructed for the same sample statistics. (a) 17. For c=0.88​, choose the corresponding normal confidence interval below (b) 18. For c=0.90​, choose the corresponding normal confidence interval below (c) 19. For c=0.95​, choose the corresponding normal confidence interval below (d) 20. For c=0.98​, choose the corresponding normal confidence interval

(a) a (57.6, 61.4) (b) a (57.5, 61.5) (c) b (57.1, 61.9) (d) d (56.7, 62.3) *(57.6, 61.4)=3.8 **(57.5, 61.5)=4 ***(57.1, 61.9)=4.8 ****(56.7, 62.3)=5.6

ANSWER 11 (c) c. With​ 90% confidence, it can be said that the population mean price lies in the first interval. With​ 95% confidence, it can be said that the population mean price lies in the second interval. The​ 95% confidence interval is wider than the​ 90% 90=121.59-114.41=7.18 95=122.28-113.72=8.56

QUESTION 11 (c) Interpret the results. Choose the correct answer below. a. With​ 90% confidence, it can be said that the population mean price lies in the first interval. With​ 95% confidence, it can be said that the population mean price lies in the second interval. The​ 95% confidence interval is narrower than the​ 90% b. With​ 90% confidence, it can be said that the sample mean price lies in the first interval. With​ 95% confidence, it can be said that the sample mean price lies in the second interval. The​ 95% confidence interval is wider than the​ 90% c. With​ 90% confidence, it can be said that the population mean price lies in the first interval. With​ 95% confidence, it can be said that the population mean price lies in the second interval. The​ 95% confidence interval is wider than the​ 90%

ANSWER 11 (a) 114.41; 121.59 90c=1.645 (1.645x16.20)/*square root*55=3.59 118-3.59=114.41 118+3.59=121.59 (b) 113.72; 122.28 95c=1.96 (1.96x16.20)/*square root*55=4.28 118-4.28=113.72 118+4.28=122.28 (c) see slide 12

QUESTION 11 You are given the sample mean and the population standard deviation. Use this information to construct the​ 90% and​ 95% confidence intervals for the population mean. Interpret the results and compare the widths of the confidence intervals. If​ convenient, use technology to construct the confidence intervals. A random sample of 55 home theater systems has a mean price of ​$118.00. Assume the population standard deviation is ​$16.20. (a) Construct a​ 90% confidence interval for the population mean. The​ 90% confidence interval is (_, _) (b) Construct a​ 95% confidence interval for the population mean. The 95% confidence interval is (_, _) (c) see slide 12

ANSWER 12 (a) 107.93; 114.09 90c=1.645 (1.645x10.90)/*square root*34=3.08 111.01-3.08=107.93 111.01+3.08=114.09 (b) 107.35; 114.67 95c=1.96 (1.96x10.90)/*square root*34=3.66 111.01-3.66=107.35 111.01+3.66=114.67 (c) a. The 95% confidence interval 114.09-107.93=6.16 114.67-107.35=7.32 (d) see slide 14

QUESTION 12 You are given the sample mean and the population standard deviation. Use this information to construct the​ 90% and​ 95% confidence intervals for the population mean. Interpret the results and compare the widths of the confidence intervals. From a random sample of 34 business​ days, the mean closing price of a certain stock was ​$111.01. Assume the population standard deviation is ​$10.90. (a) The 90% confidence interval is (_, _) (b) The 95% confidence interval is (_, _) (c) Which interval is wider? Choose the correct answer below. a. The 95% confidence interval b. The 90% confidence interval (d) see slide 14

ANSWER 12 (d) d. You can be​ 90% confident that the population mean price of the stock is between the bounds of the​ 90% confidence​ interval, and​ 95% confident for the​ 95% interval

QUESTION 12 Interpret the results. a. You can be certain that the population mean price of the stock is either between the lower bounds of the​ 90% and​ 95% confidence intervals or the upper bounds of the​ 90% and​ 95% confidence intervals b. You can be​ 90% confident that the population mean price of the stock is outside the bounds of the​ 90% confidence​ interval, and​ 95% confident for the​ 95% interval c. You can be certain that the closing price of the stock was within the​ 90% confidence interval for approximately 31 of the 34 ​days, and was within the​ 95% confidence interval for approximately 32 of the 34 days d. You can be​ 90% confident that the population mean price of the stock is between the bounds of the​ 90% confidence​ interval, and​ 95% confident for the​ 95% interval

ANSWER 13 c. It would be very unlikely because the margin of error is large enough that the odds of selecting an exact value is very low 0.90=0.8159 E=(0.8159)(200.10/*square root*51)=22.86

QUESTION 13 From a random sample of 51 business​ days, gold prices had a mean of $1352.17. Assume the population standard deviation is $200.10. This creates a 90% confidence interval for the population mean of ($1306.08, $1398.26). Does it seem possible that the population mean could equal the sample mean​ exactly? Explain. Choose the correct answer below. a. It is possible because the margin of error is small enough that the odds of selecting an exact value is not unlikely b. It is not possible because the sample mean is inside of the confidence interval c. It would be very unlikely because the margin of error is large enough that the odds of selecting an exact value is very low d. It is possible because the sample mean is outside of the confidence interval

ANSWER 14 (a) will widen (b) will narrow (c) will widen

QUESTION 14 If all other quantities remain the​ same, how does the indicated change affect the width of a confidence​ interval? ​(a) Increase in the level of confidence ​(b) Increase in the sample size ​(c) Increase in the population standard deviation (a) How does an increase in the level of confidence affect the width of a confidence​ interval? Choose the correct answer below. An increase in the level of confidence _ the confidence interval (b) How does an increase in the sample size affect the width of a confidence​ interval? Choose the correct answer below. An increase in the sample size _ the confidence interval (c) How does an increase in the population standard deviation affect the width of a confidence​ interval? Choose the correct answer below. An increase in the population standard deviation _ the confidence interval

ANSWER 15 (a) 65 (1.96x3.11/0.76)^2=64.3=65 (b) see slide 18

QUESTION 15 A cheese processing company wants to estimate the mean cholesterol content of all​ one-ounce servings of a type of cheese. The estimate must be within 0.76 milligram of the population mean. ​(a) Determine the minimum sample size required to construct a 95​% confidence interval for the population mean. Assume the population standard deviation is 3.11 milligrams. ​(b) The sample mean is 27 milligrams. Using the minimum sample size with a 95​% level of​ confidence, does it seem likely that the population mean could be within 3​% of the sample​ mean? within 0.3​% of the sample​ mean? Explain. (a) The minimum sample size required to construct a 95​% confidence interval is _ servings (b) see slide 18

ANSWER 15 (b) 26.24; 27.76; seems; entirely contains; does not seem; overlaps but does not entirely contain 1-0.95=0.05 0.05/2=0.025 0.025=1.96 3.11/*square root*65=0.3857 0.3857x1.96=0.7561 27-0.7561=26.2439 27+0.7561=27.7561 27x0.03=0.81 27x0.3=8.1

QUESTION 15 (b) (b) The 95% confidence interval is (_, _). Its _ likely that the population mean could be within 3% of the sample mean because the interval formed by the values 3% away from the sample mean _ the confidence interval. It _ seem likely that the population mean could be within 0.3% of the sample mean because the interval formed by the values of 0.3% away from the sample mean _ the confidence interval.

Which of these is the most unbiased point estimate of the population mean μ​? a. Sample range b. Sample mean​ x̄ c. Sample median d. Sample standard deviation s e. Sample mode f. Sample variation s^2

b. Sample mean​ x̄