chapter 9 - statistics
Compute the critical value zα/2 that corresponds to a 91% level of confidence.
1.7 (1-0.91)*100% = 0.09 0.09/2 = 0.045 z0.045 = -1.7 |-1.7| = 1.7
By how many times does the sample size have to be increased to decrease the margin of error by a factor of 1/9?
9^2 = 81 Increasing the sample size by a factor M results in the margin of error decreasing by a factor of 1M.
What happens to the interval as the level of confidence is changed? Explain why this is a logical result.
As the level of confidence increases, the width of the interval increases. This makes sense since the margin of error increases as well.
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 4 points with 99% confidence assuming s=12.8 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 68 subjects. A 90% confidence level requires 28 subjects. Decreasing the confidence level decreases the sample size needed.
We are 95% confident that the mean number of hours worked by adults in this country in the previous week was between 41.5 hours and 43.9 hours.
Correct. This interpretation is reasonable.
95% of adults in this country worked between 41.5 hours and 43.9 hours last week.
Flawed. The interpretation should be about the mean number of hours worked by adults in the whole country, not about adults in the particular area.
There is a 95% chance the mean number of hours worked by adults in this country in the previous week was between 41.5 hours and 43.9 hours.
Flawed. This interpretation implies that the population mean varies rather than the interval.
Explain what "90% confidence" means in a 90% confidence interval.
If 100 different confidence intervals are constructed, each based on a different sample of size n from the same population, then we expect 90 of the intervals to include the parameter and 10 to not include the parameter.
In a survey conducted by the Gallup Organization, 1100 adult Americans were asked how many hours they worked in the previous week. Based on the results, a 95% confidence interval for the mean number of hours worked had a lower bound of 42.7 and an upper bound of 44.5. Provide two recommendations for decreasing the margin of error of the interval.
Increase the sample size. Decrease the confidence level.
Construct a 90% confidence interval of the population proportion using the given information. x=40, n=200
LB = 0.153 UB = 0.247
Two researchers, Jaime and Mariya, are each constructing confidence intervals for the proportion of a population who is left-handed. They find the point estimate is 0.13. Each independently constructed a confidence interval based on the point estimate, but Jaime's interval has a lower bound of 0.118 and an upper bound of 0.142, while Mariya's interval has a lower bound of 0.087 and an upper bound of 0.156. Which interval is wrong? Why?
Mariya's interval is wrong because it is not centered on the point estimate.
A trade magazine routinely checks the drive-through service times of fast-food restaurants. An 80% confidence interval that results from examining 601 customers in one fast-food chain's drive-through has a lower bound of 177.9 seconds and an upper bound of 181.5 seconds. What does this mean?
One can be 80% confident that the mean drive-through service time of this fast-food chain is between 177.9 seconds and 181.5 seconds.
The data from a simple random sample with 25 observations was used to construct the plots given below. The normal probability plot that was constructed has a correlation coefficient of 0.950. Judge whether a t-interval could be constructed using the data in the sample.
The normal probability plot does not suggest the data could come from a normal population because 0.950 < 0.959 and the boxplot shows outliers, so a t-interval could not be constructed.
We are 93% to 97% confident 55% of adults in the country during the period of economic uncertainty felt wages paid to workers in industry were too low. Is the interpretation reasonable?
The interpretation is flawed. The interpretation indicates that the level of confidence is varying.
We are 95% confident 55% of adults in the country during the period of economic uncertainty felt wages paid to workers in industry were too low. Is the interpretation reasonable?
The interpretation is flawed. The interpretation provides no interval about the population proportion.
In 95% of samples of adults in the country during the period of economic uncertainty, the proportion who believed wages paid to workers in industry were too low is between 0.53 and 0.57. Is the interpretation reasonable?
The interpretation is flawed. The interpretation suggests that this interval sets the standard for all the other intervals, which is not true.
We are 95% confident that the interval from 0.53 to 0.57 contains the true proportion of adults in the country during the period of economic uncertainty who believed wages paid to workers in industry were too low. Is the interpretation reasonable?
The interpretation is reasonable.
A group conducted a poll of 2016 likely voters just prior to an election. The results of the survey indicated that candidate A would receive 46% of the popular vote and candidate B would receive 42% of the popular vote. The margin of error was reported to be 4%. The group reported that the race was too close to call. Use the concept of a confidence interval to explain what this means.
The margin of error suggests candidate A may receive between 42% and 50% of the popular vote and candidate B may receive between 38% and 46% of the popular vote. Because the poll estimates overlap when accounting for margin of error, the poll cannot predict the winner.
The data from a simple random sample with 25 observations was used to construct the plots given below. The normal probability plot that was constructed has a correlation coefficient of 0.948. Judge whether a t-interval could be constructed using the data in the sample.
The normal probability plot does not suggest the data could come from a normal population because 0.948 <0.959 and the boxplot shows outliers, so a t-interval could not be constructed.
The data from a simple random sample with 25 observations was used to construct the plots given below. The normal probability plot that was constructed has a correlation coefficient of 0.981. Judge whether a t-interval could be constructed using the data in the sample.
The normal probability plot suggests the data could come from a normal population because 0.981 > 0.959 and the boxplot does not show outliers, so a t-interval could be constructed.
Determine the point estimate of the population mean and margin of error for the confidence interval. Lower bound is 18, upper bound is 24.
The point estimate of the population mean is 21. The margin of error for the confidence interval is 3.
What would you recommend to a researcher who wants to increase the precision of the interval, but does not have access to additional data?
The researcher could decrease the level of confidence.
To construct a confidence interval about the mean, the population from which the sample is drawn must be approximately normal.
false
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 0.05 with 95% confidence if (a) she uses a previous estimate of 0.54? (b) she does not use any prior estimates?
a. 382 b. 384
A nutritionist wants to determine how much time nationally people spend eating and drinking. Suppose for a random sample of 932 people age 15 or older, the mean amount of time spent eating or drinking per day is 1.44 hours with a standard deviation of 0.73 hour. Complete parts (a) through (d) below.
a. Since the distribution of time spent eating and drinking each day is not normally distributed (skewed right), the sample must be large so that the distribution of the sample mean will be approximately normal. b. The sample size is less than 5% of the population. c. The nutritionist is 95% confident that the mean amount of time spent eating or drinking per day is between 1.393 and 1.487 hours. d. No; the interval is about people age 15 or older. The mean amount of time spent eating or drinking per day for 9-year-olds may differ.
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)*100%
A ________ ________ is the value of a statistic that estimates the value of a parameter.
point estimate
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=0.552, upper bound=0.858, n=1200
point estimate = (ub+lb)/2 = 0.705 margin of error = (ub-lb)/2 = 0.153 # of individuals in the sample with the specified characteristic = n*p = 846
Determine the point estimate of the population mean and margin of error for the confidence interval. Lower bound is 23, upper bound is 27.
point estimate = 25, (23+27)/2 = 25 margin of error = 2, 27-25 = 2
A survey of 2316 adults in a certain large country aged 18 and older conducted by a reputable polling organization found that 418 have donated blood in the past two years. Complete parts (a) through (c) below.
point estimate = 418/2316 = 0.18 The sample can be assumed to be a simple random sample, the value of np(1-p) is 341.841, which is greater than or equal to 10, and the sample size can be assumed to be less than or equal to 5% of the population size. We are 90% confident the proportion of adults in the country aged 18 and older who have donated blood in the past two years is between 0.167 and 0.194.
The procedure for constructing a confidence interval about a mean is _______, which means minor departures from normality do not affect the accuracy of the interval.
robust