Chapter 8 & 9 Test

True

True or False: The distribution of the sample​ mean will be normally distributed if the sample is obtained from a population that is normally​ distributed, regardless of the sample size.

DECREASE THE CONFIDENCE LEVEL AND INCREASE THE SAMPLE SIZE

n a survey conducted by a polling​ company, 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.

KNOW THIS

​P(90 less than or equals ≤ x overbar x less than or equals ≤​110) for a random sample of size n equals =20 has a higher probability. As n​ increases, the standard deviation decreases.

Since it is within the margin of error for candidate A to receive 43​% of the popular vote and for candidate B to receive 49​%, or for candidate A to receive 51​% of the popular vote and for candidate B to receive 41​%, the poll cannot predict the winner.

A group conducted a poll of 2025 likely voters just prior to an election. The results of the survey indicated that candidate A would receive 47​% of the popular vote and candidate B would receive 45​% 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.

There is 90​% confidence that the proportion of the adult citizens of the nation that dreaded​ Valentine's Day is between 0.099 and 0.201

A national survey of 1500 adult citizens of a nation found that 15​% dreaded​ Valentine's Day. The margin of error for the survey was 5.1 percentage points with 90​% confidence. Explain what this means.

If 100 different confidence intervals are​ constructed, each based on a different sample of size n from the same​ population, then we expect 98 of the intervals to include the parameter and 2 to not include the parameter.

Explain what ​" 98​% ​confidence" means in a 98​% confidence interval.

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.

Explain why the​ t-distribution has less spread as the number of degrees of freedom increases.

Matthew's estimate will have the smaller margin of error because the larger sample size more than compensates for the higher level of confidence.

Katrina wants to estimate the proportion of adults who read at least 10 books last year. To do​ so, she obtains a simple random sample of 100 adults and constructs a​ 95% confidence interval. Matthew also wants to estimate the proportion of adults who read at least 10 books last year. He obtains a simple random sample of 400 adults and constructs a​ 99% confidence interval. Assuming both Katrina and Matthew obtained the same point​ estimate, whose estimate will have the smaller margin of​ error? Justify your answer.

The margin of error for Population A will be smaller because its sample standard deviation will be smaller.

Suppose you have two​ populations: Population A long dash —All students at a state college ​(N equals =​21,000) and Population B long dash —All residents of a town in the state ​(N equals =​21,000). You want to estimate the mean age of each population using two separate samples each of size n equals =75. If you construct a​ 95% confidence interval for each population​ mean, will the margin of error for population A be​ larger, the​ same, or smaller than the margin of error for population​ B? Justify your reasoning.

False. The​ chi-square distribution is skewed to the right.

TRUE OR FALSE: The​ chi-square distribution is symmetric.

FALSE

TRUE OR FALSE: To construct a confidence interval about the​ mean, the population from which the sample is drawn must be approximately normal.

Standard Error of the Mean

The standard deviation of the sampling distribution of x overbar x​, denoted sigma Subscript x overbar σx​, is called the

False. The sample size must be increased by a factor of four to cut the standard error in half.

To cut the standard error of the mean in​ half, the sample size must be doubled.

False. To construct a confidence interval about a population variance or standard​ deviation, the sample must come from a normally distributed population.

True or​ false: To construct a confidence interval about a population variance or standard​ deviation, either the population from which the sample is drawn must be​ normal, or the sample size must be large.

A point estimate

is the value of a statistic that estimates the value of a parameter.