Stat - Chapter 9
The following data represent the asking price of a simple random sample of homes for sale. Construct a 99% confidence interval with and without the outlier included. Comment on the effect the outlier has on the confidence interval. (a) Construct a 99% confidence interval with the outlier included. (b) Construct a 99% confidence interval with the outlier removed. The outlier is $459,900. (c) Comment on the effect the outlier has on the confidence interval.
(a) ($156764, $307753) Found by doing Stats > T Stats > One Sample > With Data using StatCrunch. (b) ($167295, $255832) Found by doing Stats > T Stats > One Sample > With Data using StatCrunch. (c) The outlier caused the width of the confidence interval to increase.
A trade magazine routinely checks the drive-through service times of fast-food restaurants. A 95% confidence interval that results from examining 504 customers in one fast-food chain's drive-through has a lower bound of 159.6 seconds and an upper bound of 162.6 seconds. What does this mean?
One can be 95% confident that the mean drive-through service time of this fast-food chain is between 159.6 seconds and 162.6 seconds.
Based on interviews with 99 SARS patients, researchers found that the mean incubation period was 5 days, with a standard deviation of 15.8 days. Based on this information, construct a 95% confidence interval for the mean incubation period of the SARS virus. Interpret the interval.
The lower bound is 1.85 days. The upper bound is 8.15 days. The lower and upper bounds are found by using the Stat > T Stats > One Sample > With Summary on StatCrunch. There is 95% confidence that the mean incubation period lies between the lower and upper bounds of the interval.
In a poll, 69% of the people polled answered yes to the question "Are you in favor of the death penalty for a person convicted of murder?" The margin of error in the poll was 22%, and the estimate was made with 94% confidence. At least how many people were surveyed?
The minimum number of surveyed people was 1891. Found by doing 0.69 (1-0.69) (1.88/0.02)^2, with 1.88 being the z-value for 94%.
Fill in the blanks in the statement below. The _______ represents the expected proportion of intervals that will contain the parameter if a large number of different samples of size n is obtained. It is denoted _______.
level of confidence; (1-a) x 100%
A ________ ________ is the value of a statistic that estimates the value of a parameter.
point estimate
Compute the critical value zα/2 that corresponds to a 80% level of confidence.
za/2=1.28
The following data represent the concentration of organic carbon (mg/L) collected from organic soil. (Note: x overbar=17.61 mg/L and s=7.86 mg/L) (a) Determine a point estimate for the population mean. (b) Construct a 99% confidence interval for the mean concentration of dissolved organic carbon collected from organic soil.
(a) A point estimate for the population mean is 17.61 mg/L. (b) A normal probability plot suggests the data could come from a population that is normally distributed, and a boxplot does not show any outliers. (12.58, 22.64) Found by doing Stat > T Stats > One Sample > With Data on StatCrunch.
The results of a question asked in a simple random survey are reported in the accompanying table. The question asked was "If you own a cell phone, what was your cell phone bill last month? Express answers to the penny." Use the data to complete parts (a) through (e) below. (a) Draw a histogram of the data and comment on the shape. (b) Draw a boxplot of the data. Are there any outliers? (c) Based on the results to parts (a) and (b), explain why a large sample size might be desirable to construct a confidence interval for the mean monthly cell phone bill. Choose the correct answer below. (d) Use statistical software to determine the point estimate of the mean monthly cell phone bill. (e) Use statistical software to construct a 99% confidence interval for the mean monthly cell phone bill.
(a) Histogram shown. Use StatCrunch to approximate the histogram. The histogram is skewed right. (b) Boxplot shown. Use StatCrunch to approximate the box plot. There are outliers because there are several values that fall outside 1.5(IQR) of Q1 and Q3. (c) The results to parts (a) and (b) indicate that the underlying population is non-normal. Such populations require large sample sizes to construct valid confidence intervals. (d) $95.46 (e) ($77.59, $113.32)
Determine the point estimate of the population proportion, the margin of error for the following confidence interval, and the number of individuals in the sample with the specified characteristic, x, for the sample size provided. Lower bound=0.123, upper bound=0.407, n=1200
(a) The point estimate of the population proportion is 0.265. Found by doing p=u+b/2 = 0.123+0.407/2 = 0.265 (b) The margin of error is 0.142. Found by doing u-b/2 = 0.407-0.123/2 = 0.142 (c) The number of individuals in the sample with the specified characteristic is 318. Found by doing 1200(0.265) = 318
The following data sets represent simple random samples from a population whose mean is 100. Complete parts (a) through (e) below. (a) Compute the sample mean of each data set. (b) For each data set, construct a 95% confidence interval about the population mean. (c) What impact does the sample size n have on the width of the interval? (d) Suppose that the data value 106 was accidentally recorded as 061. For each data set, construct a 95% confidence interval using the misentered data. (e) Which intervals, if any, still capture the population mean, 100? Which of the following is the concept illustrated with the misentered data?
(a) The sample mean for data set I is 98.125. The sample mean for data set II is 98.35. The sample mean for data set III is 98.4. (b) For data set I, the lower bound is 81.14 and the upper bound is 115.11. For data set II, the lower bound is 90.54 and the upper bound is 106.16. For data set III, the lower bound is 92.66 and the upper bound is 104.14. (c) As the sample size increases, the width of the interval decreases. (d) For data set I, the lower bound is 72.63 and the upper bound is 112.37. For data set II, the lower bound is 87.43 and the upper bound is 104.77. For data set III, the lower bound is 90.65 and the upper bound is 103.15. Use StatCrunch to (e) All of the sets. The procedure for constructing the confidence interval is robust. The larger the sample size, the more resistant the mean. Therefore, the confidence interval is more robust.
A television sports commentator wants to estimate the proportion of citizens who "follow professional football." Complete parts (a) through (c). (a) What sample size should be obtained if he wants to be within 4 percentage points with 95% confidence if he uses an estimate of 54% obtained from a poll? (b) What sample size should be obtained if he wants to be within 4 percentage points with 95% confidence if he does not use any prior estimates? (c) Why are the results from parts (a) and (b) so close?
(a) The sample size is 597. Found by doing 0.54(1-0.54) (1.96/0.04)^2, with 1.96 being the z-value of 95%. (b) The sample size is 601. Found by doing (1.96/0.04)^2 (0.25), with 1.96 being the z-value of 95%. (c) The results are close because 0.54(1-0.54) = 0.2484 is very close to 0.25.
A researcher wishes to estimate the proportion of adults who have high-speed Internet access. What size sample should be obtained if she wishes the estimate to be within 0.04 with 95% confidence if (a) she uses a previous estimate of 0.58? (b) she does not use any prior estimates?
(a) n = 585, found by using 0.58(1-0.58) (1.96/0.04)^2 = 585. (b) n = 600, found by using (1.96/0.04)^2 (0.25) = 600.25.
An interactive poll found that 375 of 2,372 adults aged 18 or older have at least one tattoo. (a) Obtain a point estimate for the proportion of adults who have at least one tattoo. (b) Construct a 95% confidence interval for the proportion of adults with at least one tattoo. (c) Construct a 99% confidence interval for the proportion of adults with at least one tattoo. (d) What is the effect of increasing the level of confidence on the width of the interval?
(a) p = 0.134; found by doing 303/2264 = 0.134. (b) Lower bound: 0.120 Upper bound: 0.148 (c) Lower bound: 0.115 Upper bound: 0.152 (d) Increasing the level of confidence widens the interval.
In a trial of 200 patients who received 10-mg doses of a drug daily, 36 reported headache as a side effect. Use this information to complete parts (a) through (d) below. (a) Obtain a point estimate for the population proportion of patients who received 10-mg doses of a drug daily and reported headache as a side effect. (b) Verify that the requirements for constructing a confidence interval about p are satisfied. Are the requirements for constructing a confidence satisfied? (c) Construct a 95% confidence interval for the population proportion of patients who receive the drug and report headache as a side effect. (d) Interpret the confidence interval. Which statement below best interprets the interval?
(a) p = 0.18; found by doing 36/200 = 0.18 (b) Yes, the requirements for constructing a confidence interval are satisfied. (c) The 95% confidence interval is (0.127,0.233); found via using the proportion stats on StatCrunch (d) We are 95% confident that the interval contains the true value of p.
In a poll, 408 of 1020 randomly selected adults aged 18 or older stated that they believe there is too little spending on national defense. Use this information to complete parts (a) through (e) below. (a) Obtain a point estimate for the proportion of adults aged 18 or older who feel there is too little spending on national defense. (b) Verify that the requirements for constructing a confidence interval about p are satisfied. Are the requirements for constructing a confidence satisfied? (c) Construct a 99% confidence interval for the proportion of adults aged 18 or older who believe there is too little spending on national defense. (d) Is it possible that more than 45% of adults aged 18 or older believe there is too little spending on national defense? Is it likely? (e) Use the results of part (c) to construct a 90% confidence interval for the population proportion of adults aged 18 or older who do not believe there is too little spending on national defense.
(a) p = 0.4; found by doing 408/1020 = 0.4 (b) Yes, the requirements for constructing a confidence interval are satisfied. (c) The 90% confidence interval is (0.375, 0.425); found via using the proportion stats on StatCrunch. (d) It is possible, but not likely (e) The 90% confidence interval is (0.575, 0.625).
A doctor wants to estimate the mean HDL cholesterol of all 20- to 29-year-old females. How many subjects are needed to estimate the mean HDL cholesterol within 3 points with 99% confidence assuming s=16.5 based on earlier studies? Suppose the doctor would be content with 90% confidence. How does the decrease in confidence affect the sample size required?
A 99% confidence level requires 201 subjects. Found by using (2.575 x 16.5/3)^2, with 2.575 being the z-value for 99%. A 90% confidence level requires 82 subjects. Found by using (1.645 x 16.5/3)^2, with 1.645 being the z-value for 90%. Decreasing the confidence level decreases the sample size needed.
Determine whether the following statement is true or false. To construct a confidence interval about the mean, the population from which the sample is drawn must be approximately normal.
False
Determine the point estimate of the population mean and margin of error for the confidence interval. Lower bound is 22, upper bound is 28.
The point estimate of the population mean is 25, found by doing x = 22 + 28/2 = 25. The margin of error for the confidence interval is 3, found by doing x + E = 28, 25 + E = 28, E = 3.
Explain why the t-distribution has less spread as the number of degrees of freedom increases.
The t-distribution has less spread as the degrees of freedom increase because, as n increases, s becomes closer to σ by the law of large numbers.
