Stats Exam 2

Taylor's Mill is a neighborhood comprised of 185 homes. According to the website Zillow, the average market value of the homes in this neighborhood is $418,300 with a standard deviation of $36,500. What is the standard error for a random sample of 32 homes from this neighborhood?

$5,883.76

What is the probability that the mean balance for the selected sample is below $290?

0.0058

What is the probability that the sample mean score is more than 75, the predetermined passing score?

0.0211

Professor Elderman offers his class of 36 a pizza party if the class average is above 80. What is the probability that he will have to deliver on his promise?

0.0228

According to IDC, Apple's share of the smartphone market was 40.4% during a recent quarter. A random sample of 90 smartphone users was selected. The standard error of the proportion is

0.0517

If an interval estimate is said to be constructed at the 90% confidence level, the significance level would be

0.1

What is the probability that the mean balance for the selected sample is between $302 and $304?

0.1505

Assuming that the population of the city is very large, construct the 91% confidence interval for the true proportion of people who plan to get the vaccine and provide the lower bound of the confidence interval below. (Give your solution as a decimal fraction)

0.3539

Now, assume that the town's population is 1,400. Construct the 91% confidence interval and provide the lower bound of the confidence interval below. (Give your solution as a decimal fraction)

0.3591

Exhibit 4 According to the IRS, the tax refund in the last year follows normal distribution. The average refunds was $2,500 and the standard deviation for these refunds was $100. Compute the probability that randomly selected tax return is between $2450 and $2550.

0.382925

Provide the upper bound of the confidence interval that you calculated above (i.e. 91% confidence interval for the population proportion assuming that the population of the town is 1,400). (Give your solution as a decimal fraction)

0.448

Provide the upper bound of the confidence interval that you calculated above (i.e. 91% confidence interval for the population proportion). (Give your solution as a decimal fraction)

0.4533

Assuming that the population of the checking account balances is normally distributed, what is the probability that a randomly selected account has a balance of more than $305?

0.4644

A random sample of 84 students at a university showed an average age of 22 years and a sample standard deviation of 3 years. Find the margin of error for the 94% confidence interval. (Assume that the population of students is large relative to the sample size. Round your solution to 4 decimal places)

0.6242

What is the probability Professor Elderman's class of 36 has a class average below 78?

0.8413

Find the required probability; the probability that a random sample of 37 exams has an average score below 84. (Round your solution to 4 decimal places)

0.8903

What is the probability that the mean balance for the selected sample is above $295?

0.8966

Consider a random sample of 25 tax return. Compute the probability that the sample mean (with sample size 25) is between $2450 and $2550.

0.987581

What is the probability that a class of 36 students will have an average greater than 70 on Professor Elderman's final exam?

0.9987

Compute the standard error of the mean. (Round your answer to 4 decimal places)

1.6285

A nursery sells trees of different types and heights. These trees average 60 inches in height with a standard deviation of 16 inches. Suppose that 75 pine trees are sold for planting at City Hall. What is the standard deviation for the sample mean?

1.85

Find the critical z-value for a 95% confidence interval. (Round your solution to 4 decimal places)

1.96

The percentage of companies that use social-media tools and technologies has been increasing over the past three years. Three years ago, the percentage of companies that use social-media tools and technologies was 50%. A random sample of n companies is given to you and you found that 5 of them use social-media tools and technologies. Suppose you want to construct the 95% confidence interval for the population proportion. What is the minimum sample size n that makes the confidence interval valid.

10

A simple random sample of 64 observations was taken from a large population. The sample mean and the standard deviation were determined to be 320 and 120 respectively. The estimated standard error of the mean is

15

The population standard deviation of a large population is measured 160. A sample of 16 is taken from the population. What is the variance of the sampling distribution of the mean for this sample?

1600

A simple random sample of 5 observations from a population containing 400 elements was taken, and the following values were obtained: 12 18 19 20 21 A point estimate of the mean is

18

Consider the following population which represents the number of pieces of junk mail that some individual "X" received during the month of February. 2 4 0 3 5 1 0 2 3 0 4 6 1 1 3 5 0 2 3 0 2 3 3 1 What is the mean of the sampling distribution of the mean number of junk mail for that person?

2.25

Find the critical z-value for a 97.8% confidence interval. (Round your solution to 4 decimal places)

2.2904

Find the critical t-value for a 98% confidence interval estimation if the sample size equals 11. (Round your solution to 4 decimal places)

2.7638

Find the critical t-value for a 97.8% confidence interval estimation with 7 degrees of freedom. (Round your solution to 4 decimal places)

2.9308

Random samples of size 81 are taken from an infinite population with the mean and standard deviation equal to 200 and 18, respectively. The distribution of the population is unknown. The mean of the sampling distribution of x ¯ is

200

A sample of 16 people reveals the mean yearly consumption of milk to be 45 gallons with a standard deviation of 8 gallons. Assume the population is normally distributed. What is the margin of error for the 90% confidence interval? You can consult the critical values in the table below. Critical Value df value Z.1 - 1.28 z.05 - 1.645 t.1 15 1.34 t.05 15 1.75

3.5

Construct 96% confidence interval for the population mean and provide the lower bound of the confidence interval below.

39.206

Construct 88% confidence interval for the population mean and provide the lower bound of the confidence interval below.

39.8992

Provide the upper bound of the confidence interval that you calculated above (i.e. 88% confidence interval for the population mean).

44.1008

Provide the upper bound of the confidence interval that you calculated above (i.e. 96% confidence interval for the population mean).

44.794

Compute the 95% confidence interval for the mean monthly incomes of the workers. Provide the lower bound of the interval below and give your answer in dollars.

5,129.33

In order to determine an interval for the mean of a population with unknown standard deviation, a sample of 61 items is selected. The mean of the sample is determined to be 23. The number of degrees of freedom for the critical t value is

60

Construct the 96% confidence interval for the population mean and provide the lower bound of the confidence interval below.

68.8388

A population has a standard deviation of 80. A random sample of 400 items from this population is selected. The sample mean is determined to be 200. Find the margin of error at 94% confidence level assuming that the population is large. (Round your solution to 4 decimal places)

7.5232

Assume that the information about average sales was lost. You decided to repeat the study and collect information for another sample. This time you obtain a random sample of 400 sales which accidentally resulted in the same average sale of $75.00. Construct the 96% confidence interval for the population mean and provide the lower bound of the confidence interval below.

72.5355

Provide the upper bound of the confidence interval that you calculated above (i.e. 96% confidence interval for the population mean when you collected new data).

77.4645

Provide the upper bound of the confidence interval for the mean monthly incomes of the workers that you calculated above. Give your answer in dollars.

8,204

Provide the upper bound of the confidence interval that you calculated above (i.e. 96% confidence interval for the population mean).

81.1612

What is the mean of the sampling distribution of x ¯?

82

Suppose that the average math SAT score for 5,392 full-time students enrolled at Madison College is 490.4 with a standard deviation of 63.7. A random sample of 49 students has been selected. The standard error of the mean for this sample is

9.1

Which of the following meets the requirements of a simple random sample?

A population contains 10 members under the age of 25 and 20 members over the age of 25. The sample will include six people chosen at random, without regard to age.

What is the probability that a class of 15 students will have a class average greater than 70 on Professor Elderman's final exam?

Cannot be determined

What is the probability that a randomly selected student in the class will score greater than 70 on Professor Elderman's final exam?

Cannot be determined

For air travelers, one of the biggest complaints is of the waiting time between when the airplane taxis away from the terminal until the flight takes off. This waiting time is known to have a right skewed distribution with a mean of 10 minutes and a standard deviation of 8 minutes. Suppose 100 fights have been randomly sampled. Describe the sampling distribution of the mean waiting time between when the airplane taxis away from the terminal until the flight takes off for these 100 flights.

Distribution is approximately normal with the mean = 10 minutes and the standard error = 0.8 minutes

Which formula would you use for calculating the standard error of the mean: finite or infinite population?

Finite, because n/N > 0.05

For a given confidence level and sample size, which of the following is true in the interval estimation of the population mean when σ is known?

If the population standard deviation is larger, the interval is wider.

After computing a confidence interval, the user believes the results are meaningless because the width of the interval is too large. Which one of the following is the best recommendation?

Increase the sample size

Refer to above. Which formula would you use for calculating the standard error of the proportion: finite or infinite population?

Infinite. The population size is not given and, therefore, in our class we assume that the population is large enough so that n/N is less than or equal to 0.05.

How does the variance of the sample mean compare to the variance of the population?

It is smaller and therefore suggests that averages have less variation than individual observations.

Which one of the following statements about confidence interval is true?

Larger sample size and larger significance level cause narrower confidence interval.

According to a recent Catalyst Census, 16% of executive officers were women with companies that have company headquarters in the Midwest. A random sample of 154 executive officers from these companies was selected. What is the shape of the sampling distribution of the proportion?

Normal (or can be approximated by a normal) because np ≥ 5 and nq ≥ 5.

Exhibit 2 Consider a random sample consisting of 50 customers who entered the store and 30 customers made a purchase among 50 customers. Which distribution will you use for a 90% confidence interval?

Normal distribution since np ≥ 5 and nq ≥ 5

Whenever estimating the confidence interval for the population proportion, which of the following is required?

Sample size and populations proportion should be such that n p ≥ 5 and n(1-p)>=5

What is the shape of the sampling distribution of x ¯?

Sampling distribution of x ¯ is normal because the sample size is >30 and CLT applies in this case.

Susan would like to conduct a survey of homeowners in the Meadowbrook neighborhood to get their opinions on proposed road modifications in the area. Which of the following would be a stratified sampling?

Susan ensures that her sample contains a number of two-story, split-level, and ranch homes in her sample that corresponds to the number of homes in the neighborhood.

It is known that the population is not normally distributed. Also, the population standard deviation is not known. A sample of 6 items is selected from this population to develop an interval estimate for the mean of the population (µ). Choose the correct alternative below.

The sample size must be increased for constructing the confidence interval.

Confidence intervals of the population mean may be created for the cases when the population standard deviation is known or unknown. How are these two cases treated differently?

Use the z distribution when σ is known; use the t distribution when σ is unknown.

Given that a 95% confidence interval for the population mean is (6.5, 12.5), we can state that

We expect that 95% of all possible sample means drawn from the population will produce confidence intervals that include the population's mean (which makes us 95% confident that interval (6.5, 12.5) contains the population mean)

How would you be wrong if the town's population was in fact 1,400 but you computed the confidence interval not taking this into account?

We would fail to take into account the finite population correction factor which would make the confidence interval wider.

Are there any additional assumptions needed in this problem to ensure that computed confidence interval is reliable?

Yes, the population needs to be normally distributed because the sample is too small to apply CLT.

In an examination of purchasing patterns of shoppers, a sample of 16 shoppers revealed that they spent, on average, $54 per hour of shopping with the standard deviation of $21. Assuming that the amount spent per hour of shopping is normally distributed, find a 90% confidence interval for the mean amount. Consult the following table to find the critical value: Critical Value df Value z0.1 - 1.282 z0.05 - 1.645 t0.1 15 1.34 t0.05 15 1.75

[$44.8125; $63.1875]

At an academically challenging high school, the average GPA of a high school senior is known to be normally distributed with a variance of 0.25. A sample of 20 seniors is taken and their average GPA is found to be 2.71. Assume that z 0.05 = 1.645 and z 0.1 = 1.282. The 90% confidence interval for the population mean GPA is:

[2.5261, 2.8939]

A 95% confidence interval for the population mean is determined to be between 100 and 120. If the confidence level is reduced to 90%, the confidence interval for µ

becomes narrower

As the number of degrees of freedom for the t distribution increases, the difference between the t distribution and the standard normal distribution

becomes smaller

Deanna has been hired to visit the local shopping mall to conduct a survey about the upcoming political election. She needs to select respondents at the mall and ask them questions about their voting tendencies. Suppose that Deanna decides to randomly select four stores in the mall and then approach each shopper in these stores and ask them to participate in the survey. Which of the following sampling techniques best describes Deanna's method?

cluster sampling

A sample of 225 college student are randomly selected across the United States, the average student fee is $30,000 with a sample standard deviation of $3,000. If you want to construct a 95% confidence interval for the average student fee for all college students based on this information, one would need to assume a. the population is normally distributed b. the population standard deviation is known c. the population standard size is large d. no additional assumptions is needed

d

Which one of the following statements is true? a. The decreased significance level causes wider confidence level b. The 95% confidence interval for the population mean implies that the population mean will be included in that interval with 95% probability. c. When sample size n is equal or greater than 30, sample mean x can be approximated by normal distribution by Law of Large Numbers. d. All kinds of confidence interval for the population proportion contains the sample proportion

d

In the point estimation,

data from the sample is used to estimate the population parameter

As the sample size increases, the margin of error

decreases

The value added and subtracted from a point estimate in order to develop an interval estimate of the population parameter is known as the

margin of error

In order to construct the interval estimate of the population mean when σ is known and the sample is very small, the population

must have a normal distribution

In computing the standard error of the mean, the finite population correction factor is used when

n / N > 0.05

Taylor's Mill is a neighborhood comprised of 185 homes. According to the website Zillow, the average market value of the homes in this neighborhood is $418,300 with a standard deviation of $36,500. A random sample of 32 homes from this neighborhood was selected. The sampling distribution of x ¯ is

normal

The battery life of the iPhone is normally distributed with the mean of 8.0 hours and the standard deviation 1.5 hours. A random sample of 15 iPhones is taken. The sampling distribution of the sample means for the battery life is

normal

A simple random sample of 64 observations was taken from a large population. The sample mean and the standard deviation were determined to be 320 and 120 respectively. The sampling distribution of x ¯ is

normal because the sample size is ≥30

A random sample of 130 mortgages in the state of Mississippi was randomly selected. From this sample, 14 were found to be delinquent on their current payment. What is the mean of the sampling distribution of the proportion of delinquent payments?

not possible to say because population proportion is not known

The national average price for regular gasoline in February 2012 was reported to be $3.52 per gallon with the standard deviation of $0.08. A random sample of 13 gas stations was taken. The sampling distribution of the sample mean for the gasoline price

not possible to say because the sample size it too small

The sample standard deviation is the point estimator of the

population standard deviation

Sampling distribution of x ¯ is the

probability distribution of the sample mean

The Central Limit Theorem (CLT) states that the

sample means of large-sized samples will be normally distributed regardless of the shape of their population distributions

The standard deviation of the sampling distribution of x ¯ is called the

standard error of the mean

Sampling distributions describe the distribution of

statistics

Deanna has been hired to visit the local shopping mall to conduct a survey about the upcoming political election. She needs to select respondents at the mall and ask them questions about their voting tendencies. Voting records show that 50% of the local population are registered as Republicans, 35% are registered as Democrats, and the remaining as Independents. Deanna wants to be sure that her final sample has a similar proportion of voters. Which of the following sampling techniques best describes Deanna's method?

stratified

Deanna has been hired to visit the local shopping mall to conduct a survey about the upcoming political election. She needs to select respondents at the mall and ask them questions about their voting tendencies. Deanna decides to position herself by the only entrance to the mall and select every 10th shopper entering the mall to participate. Which of the following sampling techniques best describes Deanna's method?

systematic sampling

When s is used to estimate σ, the margin of error is computed by using

t distribution

As the size of the sample increases,

the standard error of the mean becomes smaller

For the interval estimation of µ when σ is known and the sample is large, the proper distribution to use is

the standard normal distribution

The confidence intervals for the population proportion are generally based on

the z distribution

Suppose that the population standard deviation, σ is known and the sample size is given as 100. The 95% confidence interval is given as [4, 396]. What is the sample mean x and population standard deviation σ? Consult the critical values in the table below. CV value Z.05 1.645 Z.025 1.96

xbar 200 ME 196 sigma 1000

Refer to Exhibit 2. What is the critical value for a 90% confidence interval?

z0.05 = 1.64

What is the relationship between the expected value of the sample mean and the expected value of the population?

μ x ¯ = μ

What is the relationship between the standard deviation of the sample mean and the population standard deviation?

σ x ¯ = σ / sqrt(n)