Statistics Chapter 21

Suppose p̂ is 0.60 and the margin of error is 0.05. The 99% confidence interval for p is _____.

(0.55, 0.65) Because we were given the margin of error, the confidence interval is 0.60 ± 0.05.

Four ounce bags of cashews packaged in a cashew plant have a standard deviation of 0.03 ounces. Suppose a random sample of 9 bags is to be taken from production and the mean weight of the sample computed. What is the standard deviation of the sampling distribution of square root of n

0.01 ounces This is the standard deviation of the sampling distribution of x¯.

A potential candidate for President has stated that she will run for office if at least 30% of Americans voice support for her candidacy. To make her decision she draws a random sample of 500 Americans. Suppose that in fact 35% of all Americans support her candidacy. The mean of the sampling distribution of p̂ is _____.

0.35 The mean of the sampling distribution of p^ is the true population proportion.

A potential candidate for President has stated that she will run for office if at least 30% of Americans voice support for her candidacy. To make her decision she draws a random sample of 500 Americans. Suppose that in fact 35% of all Americans support her candidacy. The mean of the sampling distribution of p̂ is _____%.

35

When computing a sample size for a 90% confidence interval, we'll use a z* value that leaves _____% in each tail.

5 A 90% confidence interval leaves 5% in each tail.

In a sample of 605 college students, 512 said they make a point to practice "green" habits by recycling and purchasing low-carbon impact products when possible. The sample proportion p̂ from this sample is computed as X/n. The value for X is ______.

512 X is the number of occurrences in the sample.

In a sample of 605 college students, 512 said they make a point to practice "green" habits by recycling and purchasing low-carbon impact products when possible. The sample proportion p̂ from this sample is computed as X/n. The value for n is ______.

605 n is the sample size.

A 95% confidence interval from 25 SRSs is calculated. In the long run, _____% of the intervals from the SRS of 25 will cover the true p.

95

When an opinion poll states with 95% confidence the margin of error for the sample percentage is plus or minus 3 percentage points, this means that:

95% of all samples chosen using the same method will give a sample percent within 3 percentage points of the true population value. The confidence level tells us how often our method will yield a result where the parameter is within a margin of error from the estimate. It is how often our interval finds the correct value.

A statistician calculates the 95% confidence interval from 30 SRSs. In the long run, _____of the intervals from the SRS of 30 will cover the true p.

95%. Since the sample size is the same and the sample is randomly selected, this is the coverage.

What is a parameter?

A number that summarizes information about the entire population.

In the confidence interval formula, what does z* stand for?

Critical value for probability C For any probability C between 0 and 1, there is a number z* such that any Normal distribution has probability C within z* standard deviations of the mean.

What is statistical inference on μ?

Drawing conclusions about a population mean based on a sample mean.

Which of the following are three components that would completely specify a confidence interval?

Estimate of the population parameter, the confidence level, and the margin of error.

Which of the following is true of a 95% confidence interval calculated from 25 SRSs?

In the long run, 95% of the intervals from 25 SRSs will cover p. 95% will contain the parameter, 5% will not.

What does a sampling distribution of the sample mean tell us?

The values of the sample mean from all possible samples.

What is the purpose of a confidence interval?

To provide us with a range of reasonable values that the parameter could be This is the purpose of a confidence interval.

A population distribution has a Normal shape with mean μ = 100 and standard deviation s = 20. Consider two sampling distributions from this population distribution. Sampling distribution #1 is created from the sample means from all possible random samples of size n=16; sampling distribution #2 is created from the sample means from all possible random samples of size 10. The mean of sampling distribution #1 is _____ the mean of sampling distribution #2.

equal to

For random samples of size 7 from a population, the mean of the sample means from all possible samples is _____ the population mean.

equal to

For random samples of size n from a population, the mean of the sample means from all possible samples is _____ the population mean.

equal to The mean of all possible sample means equals the mean of the population.

A population distribution has a Normal shape with mean μ = 250 and standard deviation s = 30. Consider two sampling distributions from this population distribution. Sampling distribution #1 is created from the sample means from all possible random samples of size n = 20; sampling distribution #2 is created from the sample means from all possible random samples of size 200. The standard deviation of sampling distribution #1 is _____the standard deviation of sampling distribution #2.

greater than

If the population standard deviation were 10 instead of 5, the margin of error for the population mean would be _____.

increased

For random samples of size 100 from a population, the standard deviation of the sample means from all possible samples is _____ the population standard deviation.

less than

A confidence interval is of the form: estimate ± _____.

margin of error

Decreasing the sample size increases the _____.

margin of error

According to the central limit theorem, a sampling distribution becomes _____ as sample size increases.

more Normal According to the central limit theorem, a sampling distribution becomes more Normal as sample size increases. So, the shape of a sampling distribution is closer to Normal than the population distribution. That does not mean that the shape of the sampling distribution is approximately Normal.

Individual observations are _____ than sample means from large random samples of the same size from the same population.

more variable The standard deviation of the sampling distribution of x¯equals s/squareroot of n.

If X is the number of individuals in a sample that have the desired characteristic in a sample size of n, the sample proportion p̂ is computed with which value in the denominator?.

n A sample proportion is the fraction of individuals who would possess the desired characteristic.

Historically, returns on investments in the stock market are about 7% per year. A 7% return means a $1000 investment will be worth $1070 at the end of one year. The 7% figure is a _____.

not a proportion This is an average, not a proportion. One year might be up 20%, and another year down 6%. These two years would average a 7% return. Not all statistics given as percents are proportions!

The average ACT score of all incoming freshmen at a large private university is 27. The number 27 is a(n) _____.

parameter

μ is the symbol for the _____ mean.

population

The sample mean, x¯, _____ equals the population mean.

rarely

For random samples of size 10 from a population, the mean of the _____ from all possible samples is the population mean.

sample means

Increasing the ____ reduces the margin of error.

sample size

The purpose of a confidence interval for μ is to give a range of reasonable values for _____.

the population mean

What describes how often confidence intervals will capture the true parameter value in repeated sampling?

confidence level

The amount of caffeine consumed per day by children aged eight to twelve years old has a right-skewed distribution with mean μ = 110 mg and standard deviation s = 30 mg. The mean of the sampling distribution of the sample mean for all random samples of size 9 is _____ mg.

110 The mean of the sampling distribution of the sample mean equals the mean of the population.

The amount of caffeine consumed per day by children aged eight to twelve years old has a right-skewed distribution with mean μ = 110 mg and standard deviation s = 30 mg. What is the mean of the sampling distribution of for a random sample of size 9?

110.0 mg The mean of the sampling distribution of the sample mean equals the mean of the population.

About 13% of people in the United States are left handed. One university planned a large 150-seat auditorium lecture hall with 15% "left handed" seats. A class that filled the auditorium ran out of these seats because there were 18% left-handers in the class. The mean of the sampling distribution for the proportion of left-handers in the class should be _____%.

13

About 13% of people in the United States are left-handed. One university planned a large 250-seat auditorium lecture hall with 15% "left-handed" seats. A class that filled the auditorium ran out of these seats because there were 18% left-handers in the class. The mean of the sampling distribution for the proportion of left-handers in the class should be _____.

13%

About 13% of people in the United States are left-handed. One university planned a large 250-seat auditorium lecture hall with 16% "left-handed" seats. A class that filled the auditorium ran out of these seats because there were 19% left-handers in the class. The mean of the sampling distribution for the proportion of left-handers in the class should be ____.

13%

A population distribution has a Normal shape with mean μ = 50 and standard deviation s= 4. Consider two sampling distributions from this population distribution. Sampling distribution #1 is created from the sample means from all possible random samples of size n = 8; sampling distribution #2 is created from the sample means from all possible random samples of size 64. How do the shapes compare? Which of the following is true?

Both sampling distributions are Normal.

Consider this confidence interval interpretation: The average time XYZ Company takes to process new insurance claims is 9 to 11 days with 95% confidence. What needs to be done to fix this statement?

Nothing needs to be fixed; the interpretation is correct.

Consider this confidence interval interpretation: With 99% confidence, the average time it takes passengers of a local airline to claim their luggage is somewhere between 20.9 and 29.1 minutes. What needs to be done to fix this statement?

Nothing needs to be fixed; the interpretation is correct.

If X is the number of individuals in a sample that have the desired characteristic, the sample proportion is

P^=X/n.

The confidence interval formula for p does NOT include which of the following?

Population size As long as the population size is many times the size of the sample, we do not need to know it.

Consider this confidence interval interpretation: The average time XYZ Company takes to process new insurance claims is 9 to 11 days. What needs to be done to fix this statement?

The confidence level is either not reported or not reported correctly. The confidence level is not given. A better interpretation is, The average time a local company takes to process new insurance claims is 9 to 11 days, with 95% confidence.

A very large school district in Connecticut wanted to estimate the average SAT (for college admissions) score of this year's graduating class. The district took a simple random sample of 100 seniors and calculated the 95% confidence interval for μ as 505 to 520 points. Consider this interpretation of the confidence interval: The probability that the interval of 505 to 520 points captures the large Connecticut school district's graduating seniors' average SAT score is 0.95. What needs to be done to fix this statement?

The confidence level is either not reported or not reported correctly. The probability that the confidence interval procedure gives intervals that contain the value of the parameter in repeated sampling is 0.95. But once a sample is taken and the confidence interval computed, we cannot use the word, probability. We have to report confidence instead because there is no randomness left. A better interpretation is, With 95% confidence, the mean SAT score of all graduating students is between 505 and 520 points.

If the population standard deviation were 10 instead of 5, what effect would this new standard deviation have on margin of error for a population mean?

The margin of error would be increased.

A population distribution has a Normal shape with mean μ = 50 and standard deviation s= 4. Consider two sampling distributions from this population distribution. Sampling distribution #1 is created from the sample means from all possible random samples of size n=8; sampling distribution #2 is created from the sample means from all possible random samples of size 64. How do the means compare?

The mean of sampling distribution #1 is less than the mean of sampling distribution #2.

Which one of the following is NOT part of a confidence interval for μ?

The population size

The manufacturer of a new type of light bulb did a study to estimate its mean life. They reported that the mean life of the light bulb is 1600 hours with a margin of error of 25 hours with 95% confidence. Which of the following is an appropriate interpretation of this margin of error?

The true mean life of the light bulb differs from the estimated mean by no more than 25 hours for the middle 95% of the x¯.

Consider this confidence interval interpretation: We are 95% confident the sodium content for all beef hot dogs is between 353.2 and 449.1 mg. What needs to be done to fix this statement?

The true parameter needs to be stated correctly and in context.

If X is the number of individuals in a sample that have the desired characteristic in a sample size of n, the sample proportion p̂ is computed with which value in the numerator?

X A sample proportion is the fraction of individuals who would possess the desired characteristic.

An airline wants to know the average time it takes their passengers to claim their luggage. The time to claim luggage for this airline is known to be Normally distributed with mean μ and standard deviation s = 5 minutes. If the airline took a simple random sample of 100 passengers instead of 10 passengers, the margin of error for a 95% confidence interval would _____.

decrease Increasing the sample size while keeping the confidence level fixed will decrease the margin of error and the confidence interval width.

The number of movie tickets sold per week by locations of a particular movie theater chain has a right-skewed distribution with mean μ = 5000 tickets and standard deviation s = 2000 minutes. The shape of the sampling distribution of for samples of size n = 100 will be _____.

approximately Normal

The number of minutes used per month by prepaid subscribers for a particular cell phone provider has a right-skewed distribution with mean μ = 800 minutes and standard deviation s = 500 minutes. For samples of size n= 500, the sampling distribution of the sample mean will be _____.

approximately Normal The shape of the sampling distribution of the sample mean gets closer and closer to Normal as sample size increases. So, with a sample of size 500, the shape of the sampling distribution will be approximately Normal according to the central limit theorem.