Ch. 7 AP Stats

When does the spread of the sampling distribution not depend much on the size of the population?

When the population is at least 10 times larger than the sample (n<- N/10)

The central limit theorem says that...

the sampling distribution of x bar will become approximately Normal for larger sample sizes (typically when n ≥ 30), no matter what the population shape

When the population is not Normal and the sample size is small...

the sampling distribution of x bar will resemble the population shape.

What is a sampling variability?

the value of a statistic varies in repeated random sampling

What is the mean of p hat?

(1/n)(np)= p

p hat=

(1/n)x

What is the standard deviation of p hat?

(P(1-p))/n and squared

When the 10% condition is met, the standard deviation of the sampling distribution of the sample mean is

x bar

When is a statistic used to estimate a parameter an unbiased estimator?

if the mean of its sampling distribution is equal to the true value of the parameter being estimated

What is the mean of a population/parameter?

mu

What are the rules for normality?

np>-10; n (1-p)>- 10

________ come from samples

statistics

What is a statistic?

a number that describes some characteristic of a sample

A statistic is an unbiased estimator of a parameter when...

in many samples, the values of the statistic are centered at the value of the parameter

_____ variability is better, it has a ______ distribution

low; tighter

The central limit theorem is important in stats because it allows us to use the Normal Distribution to make inferences concerning the population mean a. If the sample size is reasonably large no matter the shape of the population distribution b. If the population is Normal distributed and the sample size is reasonably large c. If the population is Normally distributed no matter the sample size d. If the population is Normally distributed and the population variance is known e. If the population size is reasonably large no matter the shape of the population distribution

a

Increasing the sample size of an opinion poll will reduce the (a) bias of the estimates made from the data collected in the poll. (b) variability of the estimates made from the data collected in the poll. (c) effect of nonresponse on the poll. (d) variability of opinions in the sample. (e) variability of opinions in the population.

b

How is a variability of a statistic described?

by the spread of its sampling distribution

The Gallup Poll has decided to increase the size of its random sample of voters from about 1500 people to about 4000 people right before an election. The poll is designed to estimate the proportion of voters who favor a new law banning smoking in public buildings. The effect of this increase is to (a) reduce the bias of the estimate. (b) increase the bias of the estimate. (c) reduce the variability of the estimate. (d) increase the variability of the estimate. (e) reduce the bias and variability of the estimate.

c

What is the sampling distribution of a statistic?

the distribution of values taken by the statistic in all possible samples of the same size from the same population

What are the 3 distinct distributions involved when we sample repeatedly and measure a variable of interest?

1) The population distribution gives the values of the variable for all the individuals in the population 2) The distribution of a sample data shows the values of the variable for all the individuals in the sample 3) The sampling distribution shows the statistic values from all the possible samples of the same size from the population

The amount that households pay service providers for access to the Internet varies quite a bit, but the mean monthly fee is $38 and the standard deviation is $10. The distribution is not Normal: many households pay a base rate for low-speed access, but some pay much more for faster connections. A sample survey asks an SRS of 500 households with Internet access how much they pay. Let x bar be the mean amount paid. Explain why you can't determine the probability that the amount a randomly selected household pays for access to the Internet exceeds $39.

Because we don't know the shape of the population distribution of monthly fees

How do you get a sampling distribution?

If you took every one of the possible samples of size n from a population, calculated the sample proportion for each, and graphed all of those values

In a residential neighborhood, the median value of a house is $200,000. For which of the following sample sizes is the sample median most likely to be above $250,000? (a) n = 10 (b) n = 50 (c) n = 100 (d) n = 1000 (e) Impossible to determine without more information.

a

Suppose you take a sample of 50 students from your school and measure their height. Which one of the following is a random variable? a. The mean of the sample data. b. The mean of the sampling distribution of mean heights for samples of size 50. c. The true mean height of all students from your school.

a

The central limit theorem is important in statistics because it allows us to use the Normal distribution to find probabilities involving the sample mean (a) if the sample size is reasonably large (for any population). (b) if the population is Normally distributed and the sample size is reasonably large. (c) if the population is Normally distributed (for any sample size). (d) if the population is Normally distributed and the population standard deviation is known (for any sample size). (e) if the population size is reasonably large (whether the population distribution is known or not).

a

The magazine Sports Illustrated asked a random sample of 750 Division I college athletes, "Do you believe performance-enhancing drugs are a problem in college sports?" Suppose that 30% of all Division I athletes think that these drugs are a problem. Let be the sample proportion who say that these drugs are a problem. Which of the following are the mean and standard deviation of the sampling distribution of the sample proportion ? (a) Mean = 0.30, SD = 0.017 (b) Mean = 0.30, SD = 0.55 (c) Mean = 0.30, SD = 0.0003 (d) Mean = 225, SD = 12.5 (e) Mean = 225, SD = 157.5

a

What is a parameter?

a number that describes some characteristic of the population

When the Large Counts condition (np ≥ 10 and n(1 − p) ≥ 10) is met, the sampling distribution of p hat will be

approximately Normal

The distribution of values taken by a statistic in all possible samples of the same size from the same population is a. The probability that the statistic is obtained b. The population parameter c. The variance of the values d. The sampling distribution of the statistic

d

Which of the following statements about the sampling distribution of the sample mean is incorrect? (a) The standard deviation of the sampling distribution will decrease as the sample size increases. (b) The standard deviation of the sampling distribution is a measure of the variability of the sample mean among repeated samples. (c) The sample mean is an unbiased estimator of the population mean. (d) The sampling distribution shows how the sample mean will vary in repeated samples. (e) The sampling distribution shows how the sample was distributed around the sample mean.

e

For a specific value of p, the standard deviation of p hat gets _______ as n gets _______

smaller; larger

________ have distributions, but ________ do not

statistics; parameters

the sample mean x bar is an unbiased estimator of...

the population mean μ

When the population is Normal...

the sampling distribution of x bar will also be Normal for any sample size.

using _____ for a sample size halves the standard deviation

4n

What is the mean of a sample/statistic?

x bar

A newborn baby has extremely low birth weight (ELBW) if it weighs less than 1000 grams. A study of the health of such children in later years examined a random sample of 219 children. Their mean weight at birth was = 810 grams. This sample mean is an unbiased estimator of the mean weight μ in the population of all ELBW babies, which means that (a) in all possible samples of size 219 from this population, the mean of the values of will equal 810. (b) in all possible samples of size 219 from this population, the mean of the values of will equal μ. (c) as we take larger and larger samples from this population, will get closer and closer to μ. (d) in all possible samples of size 219 from this population, the values of will have a distribution that is close to Normal. (e) the person measuring the children's weights does so without any error.

b

A researcher initially plans to take an SRS of size n from a population that has mean 80 and standard deviation 20. If he were to double his sample size (to 2n), the standard deviation of the sampling distribution of the sample mean would be multiplied by (a) standard deviation of 2 (b) 1/standard deviation of 2 (c) 2. (d) 1/2. (e) 1/standard déviation of 2n

b

In a statistics class of 250 students, each student is instructed to toss a coin 20 times and record the value of p hat, the sample proportion of heads. The instructor then makes a histogram of the 250 values of p hat obtained. In a second statistics class of 200 students, each student is told to toss a coin 40 times and record the value of p hat, the sample proportion of heads. The instructor then makes a histogram of the 200 values of p hat obtained. Which of the following statements regarding the two histograms of p hat-values is true? a. The first class's histogram has less spread (variability) because it is derived from a larger number of students. b. The first class's histogram has greater spread (variability) because it is derived from a smaller number of tosses per student. c.The first class's histogram is more biased because it is derived from a smaller number of tosses per student.

b

Scores on the mathematics part of the SAT exam in a recent year were roughly Normal with mean 515 and standard deviation 114. You choose an SRS of 100 students and average their SAT Math scores. Suppose that you do this many, many times. Which of the following are the mean and standard deviation of the sampling distribution of ? (a) Mean = 515, SD = 114 (b) Mean = 515, SD = (c) Mean = 515/100, SD = 114/100 (d) Mean = 515/100, SD = (e) Cannot be determined without knowing the 100 scores.

b

The student newspaper at a large university asks an SRS of 250 undergraduates, "Do you favor eliminating the carnival from the term-end celebration?" All in all, 150 of the 250 are in favor. Suppose that (unknown to you) 55% of all undergraduates favor eliminating the carnival. If you took a very large number of SRSs of size n = 250 from this population, the sampling distribution of the sample proportion p hat would be (a) exactly Normal with mean 0.55 and standard deviation 0.03. (b) approximately Normal with mean 0.55 and standard deviation 0.03. (c) exactly Normal with mean 0.60 and standard deviation 0.03. (d) approximately Normal with mean 0.60 and standard deviation 0.03. (e) heavily skewed with mean 0.55 and standard deviation 0.03.

b

Which of the following distributions has a mean that varies from sample to sample? i. The population distribution ii. The distribution of sample data iii. The sampling distribution a. i only b. ii only c. iii only d. ii and iii e. i, ii, and iii

b

A machine is designed to fill 16-ounce bottles of shampoo. When the machine is working properly, the amount poured into the bottles follows a Normal distribution with mean 16.05 ounces and standard deviation 0.1 ounce. Assume that the machine is working properly. If four bottles are randomly selected and the number of ounces in each bottle is measured, then there is about a 95% chance that the sample mean will fall in which of the following intervals? (a) 16.05 to 16.15 ounces (b) 16.00 to 16.10 ounces (c) 15.95 to 16.15 ounces (d) 15.90 to 16.20 ounces (e) 15.85 to 16.25 ounces

c

A study of voting chose 663 registered voters at random shortly after an election. Of these, 72% said they had voted in the election. Election records show that only 56% of registered voters voted in the election. Which of the following statements is true about the boldface numbers? (a) 72% is a sample; 56% is a population. (b) 72% and 56% are both statistics. (c) 72% is a statistic and 56% is a parameter. (d) 72% is a parameter and 56% is a statistic. (e) 72% and 56% are both parameters.

c

At a particular college, 78% of all students are receiving some kind of financial aid. The school newspaper selects a random sample of 100 students and 72% of the respondents say they are receiving some sort of financial aid. Which of the following is true? (a) 78% is a population and 72% is a sample. (b) 72% is a population and 78% is a sample. (c) 78% is a parameter and 72% is a statistic. (d) 72% is a parameter and 78% is a statistic. (e) 78% is a parameter and 100 is a statistic.

c

In order to use the standard deviation of x bar, which condition must be met? i. n is greater than or equal to 30 ii. The population's distribution is approx. Normal iii. The sample size is less than 10% of the pop. size a. i only b. ii only c. iii only d. iii and either i or ii e. i, ii, and iii

c

Suppose we select an SRS of size n= 100 from a large population having proportion p successes. Let p hat be the proportion of successes in the sample. For which value of p would it be safe to use the Normal approximation to the sampling distribution of p hat? a. 0.01 b. 1/11 c. 0.85 d. 0.975 e. 0.999

c

The number of hours a lightbulb burns before failing varies from bulb to bulb. The population distribution of burnout times is strongly skewed to the right. The central limit theorem says that (a) as we look at more and more bulbs, their average burnout time gets ever closer to the mean μ for all bulbs of this type. (b) the average burnout time of a large number of bulbs has a sampling distribution with the same shape (strongly skewed) as the population distribution. (c) the average burnout time of a large number of bulbs has a sampling distribution with similar shape but not as extreme (skewed, but not as strongly) as the population distribution. (d) the average burnout time of a large number of bulbs has a sampling distribution that is close to Normal. (e) the average burnout time of a large number of bulbs has a sampling distribution that is exactly Normal.

d

Suppose that you are a student aide in the library and agree to be paid according to the "random pay" system. Each week, the librarian flips a coin. If the coin comes up heads, your pay for the week is $80. If it comes up tails, your pay for the week is $40. You work for the library for 100 weeks. Suppose we choose an SRS of 2 weeks and calculate your average earnings x bar. The shape of the sampling distribution of x bar will be (a) Normal. (b) approximately Normal. (c) right-skewed. (d) left-skewed. (e) symmetric but not Normal.

e

Why is it important to check the 10% condition before calculating probabilities involving x bar? (a) To reduce the variability of the sampling distribution of . (b) To ensure that the distribution of x is approximately Normal. (c) To ensure that we can generalize the results to a larger population. (d) To ensure that will be an unbiased estimator of μ. (e) To ensure that the observations in the sample are close to independent.

e

An estimator is unbiased if...

it doesn't consistently under- or overestimate the parameter in many samples

Larger samples sizes _______ variability

lower

When the 10% condition is met, the standard deviation of the sampling distribution of the sample proportion is

p hat

_______ come from populations

parameters