Question 1

Find the mode(s)for the given sample data.
-$7.16,7.41,7.56,7.16,7.88,7.99,7.62$

Accepted Answer

The mode is the number that appears most frequently in a data set. In this case, $7.16$ appears twice, more than any other number, making it the mode.

Question 2

Solve the problem. Round your answer to two decimal places.
-Scores on a test are normally distributed with a mean of 86 and a standard deviation of 3. What is the exam score corresponding to a standard score of -0.95?

Accepted Answer

To find the exam score corresponding to a standard score (z-score) of -0.95, use the formula: score = mean + (z-score * standard deviation). Here, score = 86 + (-0.95 * 3) = 86 - 2.85 = 83.15.

Question 3

State whether you would expect the data set to be normally distributed.
-The number of siblings of the students at Bloomington High School

Accepted Answer

The number of siblings is typically a discrete variable and can be skewed depending on the population, leading to a distribution that is not normal.

Question 4

Solve the problem.
-The diameters of bolts produced by a certain machine are normally distributed with a mean of $0.30$ inches and a standard deviation of $0.01$ inches. What percentage of bolts will have a diameter greater than $0.32$ inches?

Accepted Answer

To find the percentage of bolts with a diameter greater than 0.32 inches, we calculate the z-score for 0.32 inches using the formula $z = \frac{(X - \mu)}{\sigma}$, where $X = 0.32$, $\mu = 0.30$, and $\sigma = 0.01$. This gives us $z = \frac{(0.32 - 0.30)}{0.01} = 2$. Looking up a z-score of 2 in the standard normal distribution table, we find that the area to the left of z=2 is 0.9772, meaning 97.72% of the distribution is below 0.32 inches. Therefore, the percentage above 0.32 inches is 100% - 97.72% = 2.28%.

Question 5

Find the mode(s)for the given sample data.
-Last year, nine employees of an electronics company retired. Their ages at retirement are listed below. Find the mode(s). $ \begin{array}{lll}55 & 56 & 61\end{array} $
$ \begin{array}{lll}67 & 58 & 54\end{array} $

Accepted Answer

The mode is the value that appears most frequently in a data set. Since all the ages listed occur exactly once, there is no value that appears more than once, which means there is no mode.

Question 6

For the given data value, find the standard score and the percentile.
-A data value 0.6 standard deviations above the mean.

Accepted Answer

The standard score (z-score) is a measure of how many standard deviations an element is from the mean. A data value 0.6 standard deviations above the mean has a z-score of 0.6. The percentile can be found using a z-table or a calculator, and a z-score of 0.6 corresponds to approximately the 72.57th percentile.

Question 7

Find the mode(s)for the given sample data.
-The blood types for 30 people who agreed to participate in a medical study were as follows. $\begin{array}{llllllllll}
\mathrm{O} & \mathrm{A} & \mathrm{A} & \mathrm{O} & \mathrm{A} & \mathrm{AB} & \mathrm{O} & \mathrm{B} & \mathrm{A} & \mathrm{O} \
\mathrm{A} & \mathrm{O} & \mathrm{A} & \mathrm{B} & \mathrm{O} & \mathrm{O} & \mathrm{O} & \mathrm{AB} & \mathrm{A} & \mathrm{A} \
\mathrm{A} & \mathrm{B} & \mathrm{O} & \mathrm{A} & \mathrm{A} & \mathrm{O} & \mathrm{O} & \mathrm{B} & \mathrm{O} & \mathrm{O}
\end{array}$

Find the mode of the blood types.

Accepted Answer

The mode is the value that appears most frequently in a data set. Blood type O appears 12 times, which is more than any other blood type.

Question 8

Provide an appropriate response.
-The area under the standard normal curve between 1 and 2 is equal to 0.1359. Scores on a particular aptitude test are normally distributed with a mean of 100 and a standard deviation of 10. Which of
The following are equal to 13.59%? a. The percentage of scores between 120 and 130
b. The percentage of scores between 110 and 120
c. The percentage of scores between 80 and 90
d. The percentage of scores between 90 and 120

Accepted Answer

The answer of Provide an appropriate response.
-The area under the...

Question 9

Find the standard score for the given data value.
-A data value in the 31st percentile.

Accepted Answer

The 31st percentile falls below the mean (50th percentile), indicating a negative z-score. A z-score of -0.5 corresponds to approximately the 31st percentile in a standard normal distribution.

Question 10

A hypothesis test is to be performed. State the null and alternative hypotheses.
-During one flu epidemic the proportion of adults nationwide who have come down with the flu is $8 \%$. The manufacturer of a flu vaccine claims that those who have been vaccinated are less likely to catch the flu.

Accepted Answer

The null hypothesis typically states there is no effect or no difference, so it would assert that the proportion of vaccinated individuals catching the flu is the same as the general rate (8%). The alternative hypothesis reflects the manufacturer's claim that the vaccine reduces the likelihood of catching the flu, hence the proportion should be less than 8%.

Question 11

A hypothesis test is to be performed. State the null and alternative hypotheses.
-Carter Motor Company claims that its new sedan, the Libra, will average better than 25 miles per gallon, which is the gas mileage of its competitor.

Accepted Answer

The null hypothesis typically states there is no effect or no difference, so it would be that the gas mileage is equal to the competitor's 25 mpg, not 23 mpg as stated in the options. The alternative hypothesis correctly captures the company's claim of better than 25 mpg. However, since none of the options correctly state the competitor's mileage as 25 mpg, option D is the closest to correct, assuming a typographical error in the options, with the null hypothesis stating equality (no difference) and the alternative hypothesis reflecting an improvement or difference.

Question 12

Identify the distribution as symmetric, left-skewed, or right-skewed.
-Number of siblings of adults in the U.S.

Accepted Answer

The distribution of the number of siblings of adults in the U.S. is likely right-skewed because most adults have a few siblings, but a smaller number have many siblings, creating a long tail to the right.

Question 13

A hypothesis test is to be performed. Describe the two possible outcomes of the test using the context of the givensituation.
-A consumer advocacy group believes that the mean volume of juice in a company's 16 -ounce juice bottles is actually less than 16 ounces.
The hypotheses are as follows:
Null hypothesis: mean volume $= 16$ ounces
Alternative hypothesis: mean volume $< 16$ ounces

Accepted Answer

The answer of A hypothesis test is to be performed....

Question 14

Find the standard deviation for the given data. Round your answer to one more decimal place than the original data.
-$20.9,10.2,26.2,48.8,20.0,24.9$

Accepted Answer

The standard deviation is calculated by finding the square root of the variance. First, find the mean of the data: $(20.9 + 10.2 + 26.2 + 48.8 + 20.0 + 24.9) / 6 = 150.0 / 6 = 25.0$. Then, calculate each data point's deviation from the mean, square those deviations, and find the average of those squared deviations (variance). Finally, take the square root of the variance to find the standard deviation. The correct calculation leads to a standard deviation of approximately 12.87, not the exact numbers for variance or the process, but the correct answer is indeed 12.87 when following the steps for standard deviation calculation.

Question 15

Find the standard score for the given data value.
-A data value in the 16th percentile.

Accepted Answer

The 16th percentile falls below the mean in a standard normal distribution, which means the z-score should be negative. A z-score of -1.0 approximately corresponds to the 16th percentile, making option C correct.

Question 16

Use the 68-95-99.7 rule to solve the problem.
-The annual precipitation for one city is normally distributed with a mean of 27.5 inches and a standard deviation of 3.2 inches. Fill in the blanks. In $99.7 \%$ of the years, the precipitation in this city is between __ and inches.

Accepted Answer

The 68-95-99.7 rule, also known as the empirical rule, states that for a normal distribution, approximately 99.7% of the data falls within three standard deviations of the mean. Therefore, to find the range of precipitation in 99.7% of the years, we calculate the mean minus three times the standard deviation and the mean plus three times the standard deviation: $27.5 - 3(3.2)$ and $27.5 + 3(3.2)$, which gives us $17.9$ and $37.1$ inches, respectively.

Question 17

Use the range rule of thumb to approximate the standard deviation.
-The race speeds for the top eight cars in a 200 -mile race are listed below. Round results to the nearest tenth.

Accepted Answer

The range rule of thumb for estimating the standard deviation is to divide the range of the data set by 4. The range is the difference between the highest and lowest values. Without the specific speeds, we can't calculate the exact range, but following the rule, the standard deviation is approximately one-fourth of the range. Given the options, $3.4$ is the only plausible estimate based on typical race speed ranges, assuming the other options represent significantly different fractions of a typical range.

Question 18

A hypothesis test is to be performed. State the null and alternative hypotheses.
-A consumer advocacy group believes that the mean volume of juice in a company's 16-ounce juice bottles is actually less than 16 ounces.

Accepted Answer

The null hypothesis typically states that there is no effect or no difference, so it would assert that the mean volume is equal to 16 ounces. The alternative hypothesis reflects the claim being tested, in this case, that the mean volume is less than 16 ounces.

Question 19

State how many peaks you would expect for the distribution described.
-Number showing when a single die is rolled 100 times

Accepted Answer

The answer of State how many peaks you would expect...

Question 20

Identify the distribution as symmetric, left-skewed, or right-skewed.
-The amounts of tax paid by U.S. residents

Accepted Answer

The distribution of the amounts of tax paid by U.S. residents is typically right-skewed because a smaller number of people earn significantly higher incomes and thus pay substantially more in taxes, while the majority earn less and pay lower taxes.

Quiz 6: Putting Statistics to Work

Find the mode(s) for the given sample data. - $7.16,7.41,7.56,7.16,7.88,7.99,7.62$

Solve the problem. Round your answer to two decimal places. -Scores on a test are normally distributed with a mean of 86 and a standard deviation of 3. What is the exam score corresponding to a standard score of -0.95?

State whether you would expect the data set to be normally distributed. -The number of siblings of the students at Bloomington High School

Solve the problem. -The diameters of bolts produced by a certain machine are normally distributed with a mean of $0.30$ inches and a standard deviation of $0.01$ inches. What percentage of bolts will have a diameter greater than $0.32$ inches?

Find the mode(s) for the given sample data. -Last year, nine employees of an electronics company retired. Their ages at retirement are listed below. Find the mode(s) . $\begin{array}{lll}55 & 56 & 61\end{array}$ $\begin{array}{lll}67 & 58 & 54\end{array}$

For the given data value, find the standard score and the percentile. -A data value 0.6 standard deviations above the mean.

Find the standard score for the given data value. -A data value in the 31st percentile.

A hypothesis test is to be performed. State the null and alternative hypotheses. -During one flu epidemic the proportion of adults nationwide who have come down with the flu is $8 \%$ . The manufacturer of a flu vaccine claims that those who have been vaccinated are less likely to catch the flu.

A hypothesis test is to be performed. State the null and alternative hypotheses. -Carter Motor Company claims that its new sedan, the Libra, will average better than 25 miles per gallon, which is the gas mileage of its competitor.

Identify the distribution as symmetric, left-skewed, or right-skewed. -Number of siblings of adults in the U.S.

Find the standard deviation for the given data. Round your answer to one more decimal place than the original data. - $20.9,10.2,26.2,48.8,20.0,24.9$

Find the standard score for the given data value. -A data value in the 16th percentile.

Use the 68-95-99.7 rule to solve the problem. -The annual precipitation for one city is normally distributed with a mean of 27.5 inches and a standard deviation of 3.2 inches. Fill in the blanks. In $99.7 \%$ of the years, the precipitation in this city is between __ and inches.

Use the range rule of thumb to approximate the standard deviation. -The race speeds for the top eight cars in a 200 -mile race are listed below. Round results to the nearest tenth.

A hypothesis test is to be performed. State the null and alternative hypotheses. -A consumer advocacy group believes that the mean volume of juice in a company's 16-ounce juice bottles is actually less than 16 ounces.

State how many peaks you would expect for the distribution described. -Number showing when a single die is rolled 100 times

Identify the distribution as symmetric, left-skewed, or right-skewed. -The amounts of tax paid by U.S. residents