AnswersProbability and Statistics with Applications Honors Sem-2-FL-1210300 2024-2025Testing a Claim about a Population Mean

Testing a Claim about a Population Mean Answers

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The distribution of professional baseball player salaries has a mean of $3.2 million. An analyst believes that the mean salary for teams on the East Coast is different. The analyst randomly selects 15 baseball players from teams on the East Coast and records their annual salaries. The mean salary for the players in the sample is $3.9 million with a standard deviation of $2.1 million. Do the data provide convincing evidence at the level that the mean salary for the baseball players on the East Coast is different from $3.2 million?

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A
No, the data do not come from a random sample of players’ salaries.
B
No, the 10% condition is not met since 15 is more than 10% of the population of all baseball players.
C
No, since it is not known that the population of all players’ salaries is approximately Normal, the Normal/large sample condition is not met.
D
Yes, all conditions for inference are met.
2
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The distribution of the heights of five-year-old children has a mean of 42.5 inches. A pediatrician believes the five-year-old children in a city are taller on average. The pediatrician selects a random sample of 40 five-year-old children and measures their heights. The mean height of the sample is 44.1 inches with a standard deviation of 3.5 inches. Do the data provide convincing evidence at the level that the mean height of five-year-old children in this city is greater than 42.5 inches?

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A
Option
Option A
B
Option
Option B
C
Option
Option C
D
Option
Option D
3

The distribution of professional baseball player salaries has a mean of $3.2 million. An analyst believes that the mean salary for teams on the East Coast is different. The analyst randomly selects 30 baseball players from teams on the East Coast and records their annual salaries. The mean salary for the players in the sample is $3.9 million with a standard deviation of $2.1 million. Do the data provide convincing evidence at the level that the mean salary for the baseball players on the East Coast is different from $3.2 million?

Question illustration
A
t = 1.83 and 0.025 < P-value < 0.05
B
z = 1.83 and 0.025 < P-value < 0.05
C
t = 1.83 and 0.05 < P-value < 0.1
D
z = 1.83 and 0.05 < P-value < 0.1
4

The mean price of houses in the US is $383,500. A real estate agent believes the mean price of houses in a local neighborhood is less than the national mean. The agent takes a random sample of 30 houses and finds the mean price to be $295,089 with a standard deviation of $156,321. Do the data provide convincing evidence at the level that the mean price of the houses in the area is less than $383,500?What are the test statistic and P-value for this significance test?Find the t-table here and the z-table here.

Question illustration
A
t = 3.10 and 0.001 < P-value < 0.0025
B
z = 3.10 and 0.001 < P-value < 0.0025
C
t = –3.10 and 0.001 < P-value < 0.0025
D
z = –3.10 and 0.001 < P-value < 0.0025
5

The distribution of the heights of five-year-old children has a mean of 42.5 inches. A pediatrician believes the five-year-old children in a city are different. The pediatrician selects a random sample of 40 five-year-old children and measures their heights. The mean height of the sample is 41.7 inches with a standard deviation of 3.3 inches. A significance test at an alpha level of produces a P-value of 0.133. What is the correct interpretation of the P-value?

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A
There is a 13.3% chance that a sample mean at least as extreme as 41.7 inches will occur by chance if the true mean height of five-year-old children is 42.5 inches.
B
Assuming the true mean height of five-year-old children is 42.5 inches, there is a 13.3% probability that the null hypothesis is true by chance alone.
C
There is a 13.3% probability that a sample mean of 41.7 inches will occur by chance alone if the true mean height of five-year-old children is 42.5 inches.
D
Assuming the true mean height of five-year-old children is 42.5 inches, there is an 86.7% probability that a sample mean height of 42.5 inches will occur by chance alone.
6

The distribution of the heights of five-year-old children has a mean of 42.5 inches. A pediatrician believes the five-year-old children in a city are taller on average. The pediatrician selects a random sample of 30 five-year-old children and measures their heights. The mean height of the sample is 43.6 inches with a standard deviation of 3.6 inches. The pediatrician conducts a one-sample t-test for and calculates a P-value of 0.052.

Question illustration
A
The pediatrician should reject the null hypothesis since 0.052 > 0.01. There is convincing evidence that the mean height of five-year-old children in this city is greater than 42.5 inches.
B
The pediatrician should accept the null hypothesis since 0.052 > 0.01. There is not convincing evidence that the mean height of five-year-old children in this city is greater than 42.5 inches.
C
The pediatrician should fail to reject the null hypothesis since 0.052 > 0.01. There is convincing evidence that the mean height of five-year-old children in this city is greater than 42.5 inches.
D
The pediatrician should fail to reject the null hypothesis since 0.052 > 0.01. There is not convincing evidence that the mean height of five-year old children in this city is greater than 42.5 inches.
7

The distribution of the heights of five-year-old children has a mean of 42.5 inches. A pediatrician believes the five-year-old children in a city are taller on average. The pediatrician selects a random sample of 40 five-year-old children and measures their heights. The mean height of the sample is 44.1 inches with a standard deviation of 3.5 inches. Do the data provide convincing evidence at the level that the mean height of five-year-old children in this city is greater than 42.5 inches?

Question illustration
A
t = negative 2.89
Option A
B
z = negative 2.89
Option B
C
t = 2.89
Option C
D
z = 2.89
Option D
8

From previous experience, the owner of an apple orchard knows that the mean weight of Gala apples is 140 grams. There has been more precipitation than usual this year. The owner believes the weights of the apples will be heavier than usual, and therefore the crop will be more profitable. The owner takes a random sample of 30 apples and records their mean weight. What is a Type I error in this situation?

A
Based on the sample mean, the owner concludes that the mean weight of apples is greater than 140 grams when the true mean weight is not greater than 140 grams.
B
Based on the sample mean, the owner concludes that the mean weight of apples is not greater than 140 grams when the true mean weight is not greater than 140 grams.
C
Based on the sample mean, the owner concludes that the mean weight of apples is greater than 140 grams when the true mean weight of the apples is greater than 140 grams.
D
Based on the sample mean, the owner concludes that the mean weight of apples is not greater than 140 grams when the true mean weight of the apples is greater than 140 grams.
9

The distribution of professional baseball player salaries has a mean of $3.2 million. An analyst believes that the mean salary for teams on the East Coast is different. The analyst randomly selects 30 baseball players from teams on the East Coast and records their annual salaries. The mean salary for the players in the sample is $3.9 million with a standard deviation of $2.1 million. Do the data provide convincing evidence at the level that the mean salary for the baseball players on the East Coast is different from $3.2 million?

Question illustration
A
Option
Option A
B
Option
Option B
C
Option
Option C
D
Option
Option D
10

The mean price of houses in the US is $383,500. A real estate agent believes the mean price of houses in a local neighborhood is less than the national mean. The agent takes a random sample of 30 houses and finds the mean price to be $295,089 with a standard deviation of $156,321. The real estate agent conducts a significance test with the alpha level for the mean price of houses in the neighborhood being less than $383,500. The P-value for this significance test is 0.002. What is the correct interpretation of the P-value?

Question illustration
A
Assuming the true mean price of houses is $383,500, there is a 0.2% probability that the null hypothesis is true by chance alone.
B
Assuming the true mean price of houses is $383,500, there is a 0.2% probability of getting a sample mean of $295,089 by chance alone.
C
Assuming the true mean price of houses is $383,500, there is a 0.2% probability of getting a sample mean at least as extreme as $295,089 by chance alone.
D
Assuming the true mean price of houses is $383,500, there is a 99.8% probability that a sample mean of $295,089 or greater will occur by chance alone.

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