Question 1

Suppose that scores for men on an aptitude test have greater variability than scores for women on the same test.In other words,the population standard deviation is greater for the population of men than for the population of women.Based on a sample of size 50,a 95% confidence interval for the mean score,μ,of all women has a margin of error of 2.2.Which of the following confidence intervals will have a smaller margin of error?
 I.A 99% confidence interval for the mean score of women.Sample size = 50
II.A 95% confidence interval for the mean score of women.Sample size = 100
III.A 95% confidence interval for the mean score of men.Sample size = 50

Accepted Answer

Increasing the sample size to 100 (as in II) reduces the margin of error, while increasing the confidence level to 99% (as in I) increases it. The variability of men's scores (as in III) does not affect the margin of error for women's scores.

Question 2

How much fat do reduced-fat cookies typically have? You take a random sample of 51 reduced-fat cookies and test them in a lab,constructing the following confidence interval: t-Interval for μ: with 90.00% Confidence,
2.3 < μ(grams of fat)< 3.4

Accepted Answer

The confidence interval is 2.3 < μ(grams of fat)< 3.4. The midpoint of the confidence interval is (2.3 + 3.4) / 2 = 2.85. The margin of error is (3.4 - 2.3) / 2 = 0.55. Therefore, the typical amount of fat in reduced-fat cookies is 2.85 grams, with a margin of error of 0.55 grams.

Question 3

How many unpopped kernels are left when you pop a bag of microwave popcorn? Quality control personnel at Yummy Popcorn take a random sample of 50 bags of popcorn.They pop each bag in a microwave and then count the number of unpopped kernels.The following interval is produced: t-interval for μ: with 99% Confidence,
11 < μ(unpopped)< 25

Accepted Answer

The answer of How many unpopped kernels are left when...

Question 4

How much fat do reduced fat cookies typically have? You take a random sample of 51 reduced-fat cookies and test them in a lab,constructing the following confidence interval: t-Interval for μ: with 90.00% Confidence,
2.3 < μ(grams of fat)< 3.4

Accepted Answer

The answer of How much fat do reduced fat cookies...

Question 5

You want to estimate the average gas price in your city for a litre of regular unleaded gasoline.From your sample of 15 gas stations,you calculate a 95% confidence interval of ($1.34,$1.40)

Accepted Answer

A 95% confidence interval means that there is a 95% probability that the true population parameter (average gas price in this case) is within the interval. So, statement A is the correct interpretation of the confidence interval. Statements B-E are incorrect.

Question 6

How tall is your average statistics classmate? To determine this,you measure the height of a random sample of 15 of your 100 fellow students,finding a 95% confidence interval for the mean height of 164.76 to 170.90 cm.

Accepted Answer

The answer of How tall is your average statistics classmate?...

Question 7

A random sample of clients at a weight loss centre were given a dietary supplement to see if it would promote weight loss.The centre reported that the 20 clients lost an average of 15.3 kg,and that a 95% confidence interval for the mean weight loss this supplement produced has a margin of error of ±4.5 kg.

Accepted Answer

The statement "We are 95% confident that the mean weight loss produced by the supplement in weight loss centre clients is between 10.8 and 19.8 kg" correctly reflects the meaning of a 95% confidence interval. It means that if we were to repeat the study and construct 100 different intervals in the same way, approximately 95 of those intervals would include the true average weight loss in the population. The other answer choices either make incorrect statements about the population or imply a level of certainty that is not supported by the evidence.

Question 8

Alarmed at the rising gas prices in your town,you decide to estimate the average gas price for a litre of regular unleaded gasoline.From your sample of 25 gas stations,you calculate a 90% confidence interval of ($1.33,$1.41).

Accepted Answer

The width of the confidence interval is $1.41 - $1.33 = $0.08.

Question 9

Suppose you have obtained a confidence interval for μ,but wish to obtain a greater degree of precision.Which of the following would result in a narrower confidence interval? 
I.Increasing the sample size while keeping the confidence level fixed
II.Decreasing the sample size while keeping the confidence level fixed
III.Increasing the confidence level while keeping the sample size fixed
IV.Decreasing the confidence level while keeping the sample size fixed

Accepted Answer

Increasing the sample size (I) decreases the standard error, which narrows the confidence interval. Decreasing the confidence level (IV) reduces the critical value (z or t), which also narrows the interval.

Question 10

Based on a sample of 32 randomly selected years,a 90% confidence interval for the mean annual precipitation in one city is from $119.1 \mathrm {~cm}$ to $125.9 \mathrm {~cm}$ .

Accepted Answer

The confidence interval given is (11.2, 12.6).

The formula for the margin of error at 90% confidence level is

z* (population standard deviation/ sqrt(n))

Since the standard deviation is not given, we can use the range formula to estimate it. The range is

12.6 - 11.2 = 1.4

Assuming a normal distribution, 1.4/4 ~ 0.35 is the estimate of the standard deviation.

So, the margin of error is

1.645*0.35/sqrt(32) ~ 0.3

Thus, the estimate of the mean precipitation is

(11.2 + 12.6)/2 = 11.9

And the margin of error is +/- 0.3

Therefore, the best choice is C)3.4 cm as the margin of error is around 3.4 cm.

Question 11

Based on a sample of size 42,a 95% confidence interval for the mean score of all students on an aptitude test is from 57.2 to 64.8.

Accepted Answer

The confidence interval is 64.8 - 57.2 = 7.6. The margin of error is half of this, or 3.8.

Question 12

A researcher wants to estimate the mean cholesterol level of people in his city.A random sample of 21 people yields an average cholesterol level of 219,with a margin of error of ±12.Assume the researcher used a confidence level of 90%.

Accepted Answer

The margin of error of ±12 indicates that the true mean cholesterol level in the city could be either 12 units higher or 12 units lower than the sample mean of 219. Since the confidence level is 90%, we can say that we are 90% confident that the true mean lies within the interval [219-12, 219+12], which simplifies to (207, 231). Thus, the correct statement is that the researcher can be 90% confident that the mean cholesterol level for people in his city is between 207 and 231. The other choices are incorrect because they either make incorrect statements about the sample, the population, or the confidence interval.

Question 13

To determine the mean number of unpopped kernels in your favorite brand of microwave popcorn,you pop a random sample of 50 bags of popcorn and construct of 98% confidence interval of (12.5,15.4).

Accepted Answer

The answer of To determine the mean number of unpopped...

Question 14

A scientist in Smallville tested the cholesterol of a random sample of 35 town residents.He constructed the following confidence interval: t-interval for μ: with 99.00% Confidence,
188 < μ(Cholesterol)< 206

Accepted Answer

The answer of A scientist in Smallville tested the cholesterol...

Question 15

How tall is the average student on your campus? To determine this,you measure the height,in centimetres,of a random sample of 9 of your fellow 100 students and calculate the following confidence interval with 95% confidence: 155.72 < μ(height)< 181.20

Accepted Answer

The confidence interval indicates that there is a 95% probability that the true population mean height of all students at the campus is between 155.72 and 181.20 cm. Option B accurately reflects this meaning. Option C is incorrect because we cannot talk about the probability of an individual classmate's height being within the interval. Option D is incorrect because height might vary across students; it is not necessary that exactly 95 out of the 100 students are within this interval. Option E incorrectly generalizes the results to all statistics students, while we had only looked at a sample from a single campus.

Question 16

How tall is your average statistics classmate? To determine this,you measure the height of a random sample of 15 of your fellow students,finding a 95% confidence interval for the mean height of 164.76 to 170.90 cm.

Accepted Answer

The 95% confidence interval for the mean height of 164.76 to 170.90 cm has a margin of error of (170.90 - 164.76)/2 = 3.07 cm. Therefore, we can say with 95% confidence that the true mean height of the population falls within 164.76 to 170.90 cm, with a margin of error of 1.43 cm (half of the width of the interval).

Question 17

A credit union took a random sample of 40 accounts and obtained the following 90% confidence interval for the mean checking account balance at the institution: $2193 < μ(balance)< $3828.

Accepted Answer

This statement correctly represents the meaning of the confidence interval. It indicates that based on this particular sample, we are 90% confident that the true mean checking account balance at the credit union is between $2193 and $3828. The other answer choices either make incorrect generalizations about the entire population or the process of creating confidence intervals.

Question 18

Data collected by child development scientists produced the following 90% confidence interval for the average age (in months)at which children say their first word: 10.6 < μ(age)< 13.1.

Accepted Answer

This choice accurately describes the meaning of a 90% confidence interval. It means that based on this sample, we can say with 90% confidence that the true population mean age at which children say their first word is between 10.6 and 13.1 months.

Question 19

Analysis of a random sample of 250 Nova Scotia nurses produced a 95% confidence interval for the mean annual salary of $$\$ 42,807 < \mu \text { (Nurse Salary) } < \$ 49,699 \text {. }$$

Accepted Answer

A confidence interval is a range of values that is likely to contain the true population parameter with a certain level of confidence. In this case, we are given a 95% confidence interval for the mean annual salary of Nova Scotia nurses based on a random sample of 250 nurses. Option D correctly states that we can be 95% confident that the true mean salary of all Nova Scotia nurses falls somewhere between $42,807 and $49,699, which is the interval obtained from the sample data. Option A and C only refer to the sample data and do not make any statement about the population parameter. Option B refers to the average nurse salary in Canada, which is outside the scope of the given sample. Option E is incorrect because the confidence interval is based on a single sample, and we cannot make a statement about the intervals from other samples.

Question 20

How much fat do reduced fat cookies typically have? You take a random sample of 51 reduced-fat cookies and test them in a lab,finding a mean fat content of 4.2 grams.You calculate a 95% confidence interval and find that the margin of error is ±0.8 grams.

Accepted Answer

The 95% confidence interval gives us a range of values that we are 95% confident the true population mean falls within. In this case, we can say with 95% confidence that the mean fat content in reduced-fat cookies is between 3.4 and 5 grams of fat, based on our sample mean of 4.2 and a margin of error of ±0.8 grams. This choice accurately reflects the interpretation of a confidence interval.

Quiz 20: Inferences About Means

How much fat do reduced-fat cookies typically have? You take a random sample of 51 reduced-fat cookies and test them in a lab,constructing the following confidence interval: t-Interval for μ: with 90.00% Confidence, 2.3 < μ(grams of fat) < 3.4

How much fat do reduced fat cookies typically have? You take a random sample of 51 reduced-fat cookies and test them in a lab,constructing the following confidence interval: t-Interval for μ: with 90.00% Confidence, 2.3 < μ(grams of fat) < 3.4

You want to estimate the average gas price in your city for a litre of regular unleaded gasoline.From your sample of 15 gas stations,you calculate a 95% confidence interval of ($1.34,$1.40)

How tall is your average statistics classmate? To determine this,you measure the height of a random sample of 15 of your 100 fellow students,finding a 95% confidence interval for the mean height of 164.76 to 170.90 cm.

Alarmed at the rising gas prices in your town,you decide to estimate the average gas price for a litre of regular unleaded gasoline.From your sample of 25 gas stations,you calculate a 90% confidence interval of ($1.33,$1.41) .

Based on a sample of 32 randomly selected years,a 90% confidence interval for the mean annual precipitation in one city is from $119.1 \mathrm {~cm}$ to $125.9 \mathrm {~cm}$ .

Based on a sample of size 42,a 95% confidence interval for the mean score of all students on an aptitude test is from 57.2 to 64.8.

A researcher wants to estimate the mean cholesterol level of people in his city.A random sample of 21 people yields an average cholesterol level of 219,with a margin of error of ±12.Assume the researcher used a confidence level of 90%.

To determine the mean number of unpopped kernels in your favorite brand of microwave popcorn,you pop a random sample of 50 bags of popcorn and construct of 98% confidence interval of (12.5,15.4) .

A scientist in Smallville tested the cholesterol of a random sample of 35 town residents.He constructed the following confidence interval: t-interval for μ: with 99.00% Confidence, 188 < μ(Cholesterol) < 206

How tall is the average student on your campus? To determine this,you measure the height,in centimetres,of a random sample of 9 of your fellow 100 students and calculate the following confidence interval with 95% confidence: 155.72 < μ(height) < 181.20

How tall is your average statistics classmate? To determine this,you measure the height of a random sample of 15 of your fellow students,finding a 95% confidence interval for the mean height of 164.76 to 170.90 cm.

A credit union took a random sample of 40 accounts and obtained the following 90% confidence interval for the mean checking account balance at the institution: $2193 < μ(balance) < $3828.

Data collected by child development scientists produced the following 90% confidence interval for the average age (in months) at which children say their first word: 10.6 < μ(age) < 13.1.

Analysis of a random sample of 250 Nova Scotia nurses produced a 95% confidence interval for the mean annual salary of $\$ 42,807 < \mu \text { (Nurse Salary) } < \$ 49,699 \text {. }$

How much fat do reduced fat cookies typically have? You take a random sample of 51 reduced-fat cookies and test them in a lab,finding a mean fat content of 4.2 grams.You calculate a 95% confidence interval and find that the margin of error is ±0.8 grams.

Quiz 20: Inferences About Means

How much fat do reduced-fat cookies typically have? You take a random sample of 51 reduced-fat cookies and test them in a lab,constructing the following confidence interval: t-Interval for μ: with 90.00% Confidence, 2.3 < μ(grams of fat) < 3.4

How much fat do reduced fat cookies typically have? You take a random sample of 51 reduced-fat cookies and test them in a lab,constructing the following confidence interval: t-Interval for μ: with 90.00% Confidence, 2.3 < μ(grams of fat) < 3.4

You want to estimate the average gas price in your city for a litre of regular unleaded gasoline.From your sample of 15 gas stations,you calculate a 95% confidence interval of ($1.34,$1.40)

How tall is your average statistics classmate? To determine this,you measure the height of a random sample of 15 of your 100 fellow students,finding a 95% confidence interval for the mean height of 164.76 to 170.90 cm.

Alarmed at the rising gas prices in your town,you decide to estimate the average gas price for a litre of regular unleaded gasoline.From your sample of 25 gas stations,you calculate a 90% confidence interval of ($1.33,$1.41) .

Based on a sample of 32 randomly selected years,a 90% confidence interval for the mean annual precipitation in one city is from 119.1 cm119.1 \mathrm {~cm}119.1 cm to 125.9 cm125.9 \mathrm {~cm}125.9 cm .

Based on a sample of size 42,a 95% confidence interval for the mean score of all students on an aptitude test is from 57.2 to 64.8.

A researcher wants to estimate the mean cholesterol level of people in his city.A random sample of 21 people yields an average cholesterol level of 219,with a margin of error of ±12.Assume the researcher used a confidence level of 90%.

To determine the mean number of unpopped kernels in your favorite brand of microwave popcorn,you pop a random sample of 50 bags of popcorn and construct of 98% confidence interval of (12.5,15.4) .

A scientist in Smallville tested the cholesterol of a random sample of 35 town residents.He constructed the following confidence interval: t-interval for μ: with 99.00% Confidence, 188 < μ(Cholesterol) < 206

How tall is the average student on your campus? To determine this,you measure the height,in centimetres,of a random sample of 9 of your fellow 100 students and calculate the following confidence interval with 95% confidence: 155.72 < μ(height) < 181.20

How tall is your average statistics classmate? To determine this,you measure the height of a random sample of 15 of your fellow students,finding a 95% confidence interval for the mean height of 164.76 to 170.90 cm.

A credit union took a random sample of 40 accounts and obtained the following 90% confidence interval for the mean checking account balance at the institution: $2193 < μ(balance) < $3828.

Data collected by child development scientists produced the following 90% confidence interval for the average age (in months) at which children say their first word: 10.6 < μ(age) < 13.1.

Analysis of a random sample of 250 Nova Scotia nurses produced a 95% confidence interval for the mean annual salary of $42,807<μ (Nurse Salary) <$49,699. \$ 42,807 < \mu \text { (Nurse Salary) } < \$ 49,699 \text {. }$42,807<μ (Nurse Salary) <$49,699.

How much fat do reduced fat cookies typically have? You take a random sample of 51 reduced-fat cookies and test them in a lab,finding a mean fat content of 4.2 grams.You calculate a 95% confidence interval and find that the margin of error is ±0.8 grams.

Based on a sample of 32 randomly selected years,a 90% confidence interval for the mean annual precipitation in one city is from $119.1 \mathrm {~cm}$ to $125.9 \mathrm {~cm}$ .

Analysis of a random sample of 250 Nova Scotia nurses produced a 95% confidence interval for the mean annual salary of $\$ 42,807 < \mu \text { (Nurse Salary) } < \$ 49,699 \text {. }$