Q: A college admits only students with SAT Reading scores in the top 10%. If the mean is 500 and the standard deviation is 100, what is the minimum score required? (Use $$z = 1.28$$ for the 90th percentile.)

$$500 + 1.28 \times 100 = 628$$

Question 1

Which of the following best describes what a z-score represents in statistics?

Accepted Answer

The number of standard deviations a value is from the mean.

Question 2

Given a dataset with mean $$\mu = 50$$ and standard deviation $$\sigma = 5$$, what is the z-score for a value of $$60$$?

Accepted Answer

$$z = \frac{60 - 50}{5} = 2$$

Question 3

If all values in a dataset are increased by 10, which of the following statements is true?

Accepted Answer

The mean increases by 10, but the standard deviation remains unchanged.

Question 4

A cereal company wants no more than 4% of its boxes to be underweight (less than 16.0 oz). If the standard deviation is $$0.2 oz$$, what mean should the machine be set to? (Use $$z = -1.75$$ for the 4th percentile.)

Accepted Answer

$$\mu = 16.0 - (-1.75)(0.2) = 16.35$$

Question 5

Which of the following is NOT a condition for using the Normal model?

Accepted Answer

The data must be categorical.

Question 6

If a dataset is rescaled by multiplying every value by 3, what happens to the mean and standard deviation?

Accepted Answer

Both the mean and standard deviation are multiplied by 3.

Question 7

A student scores 1.8 standard deviations above the mean on the SAT Math section. If the mean is 500 and the standard deviation is 100, what is the student’s score?

Accepted Answer

$$500 + 1.8 	imes 100 = 680$$

Question 8

According to the 68-95-99.7 Rule, what percentage of data falls within two standard deviations of the mean in a Normal distribution?

Accepted Answer

95%

Question 9

A company president wants to keep the mean weight of cereal boxes at 16.2 oz, but only 4% of boxes can be underweight (less than 16.0 oz). What standard deviation must the machine achieve? (Use $$z = -1.75$$ for the 4th percentile.)

Accepted Answer

$$\sigma = \frac{16.0 - 16.2}{-1.75} = 0.114$$

Question 10

Which of the following best describes the effect of standardizing a dataset (converting to z-scores)?

Accepted Answer

The mean becomes 0 and the standard deviation becomes 1.

Question 11

A Normal probability plot is used to:

Accepted Answer

Assess whether the data are approximately Normal.

Question 12

If a student’s SAT Reading score is 600, with a mean of 500 and a standard deviation of 100, what percentage of students scored lower than this student (assuming a Normal distribution)?

Accepted Answer

About 84%

Question 13

Which of the following is a correct interpretation of a z-score of $$-2$$?

Accepted Answer

The value is 2 standard deviations below the mean.

Question 14

A college admits only students with SAT Reading scores in the top 10%. If the mean is 500 and the standard deviation is 100, what is the minimum score required? (Use $$z = 1.28$$ for the 90th percentile.)

Accepted Answer

$$500 + 1.28 	imes 100 = 628$$

Question 15

Which of the following statements about the sampling distribution of the mean is TRUE?

Accepted Answer

The sampling distribution of the mean tends to be Normal, even if the underlying data are not.

Question 16

If a dataset is strongly skewed to the right, which of the following is the best course of action before applying the Normal model?

Accepted Answer

Make a histogram or Normal probability plot to check for Normality.

Question 17

A manufacturer finds that 6.7% of cereal boxes are underweight. If the company wants to reduce this to 4% without changing the mean, what must happen to the standard deviation?

Accepted Answer

The standard deviation must decrease.

Question 18

Which of the following is a correct application of the 68-95-99.7 Rule?

Accepted Answer

About 68% of data in a Normal distribution falls within one standard deviation of the mean.

Question 19

If a value has a z-score of 0, what does this indicate about the value?

Accepted Answer

It is equal to the mean.

Question 20

A student wants to be in the top quarter of ACT scores. If the ACT has a mean of 20.8 and a standard deviation of 4.8, and the top quarter on the SAT starts at a z-score of 1.40, what ACT score does the student need?

Accepted Answer

$$20.8 + 1.40 	imes 4.8 = 27.52$$

Standard Deviation as a Ruler and the Normal Model: Chapter 5 Study Notes

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