Q: Assume $$X \sim N(50, 12^2$). $$Find the 24th percentile for $$X.$$

$$X = 50 + (-0.71) \times 12 = 41.48$$

Q: A physical fitness association wants to recognize the fastest 15% of boys in a mile run. If the mean time is $$440$$ seconds and the standard deviation is $$50$$ seconds, what time must a boy beat to earn a certificate?

$$440 + (-1.04) \times 50 = 387.96$$ seconds

Q: How large a sample is needed to be 95% confident that the sample proportion of left-handed scientists will not differ from the true proportion by more than 6%, given $$p = 0.09$$?

$$n = \left( \frac{1.96^2$ \times 0.09 \times 0.91}{0.06^2$} \right) = 87$$

Q: A doctor wants to estimate the mean birth weight of infants within 2.5 ounces with 90% confidence. If $$\sigma = 5.5$$ ounces, what sample size is needed?

$$n = \left( \frac{1.645 \times 5.5}{2.5} \$$right)^2$$ = 13$$

Question 1

Which of the following properties is NOT a characteristic of a normal distribution curve?

Accepted Answer

The curve is skewed to the right.

Question 2

Given a normal distribution with mean $$\mu = 2200$$ grams and standard deviation $$\sigma = 365$$ grams, what is the $$z-$$score for a hyrax weighing $$2930$$ grams?

Accepted Answer

$$z = \frac{2930 - 2200}{365} = 2.0$$

Question 3

If the area under the normal curve to the right of $$X = 2930 is$$ $$0.0228$$, which of the following is a correct interpretation?

Accepted Answer

2.28% of hyraxes weigh more than 2930 grams.

Question 4

Find the area under the standard normal curve to the left of $$z = 1.5.$$

Accepted Answer

0.9332

Question 5

Find the area under the standard normal curve to the right of $$z = 1.$$

Accepted Answer

0.1587

Question 6

A college student takes between 5.0 and 7.5 minutes to find a parking spot. If the mean is $$6.5$$ minutes and the standard deviation is $$1$$ minute, what is the probability that a randomly selected student falls in this range?

Accepted Answer

0.7745

Question 7

A firm’s internal rate of return is normally distributed with mean $$38$$ and standard deviation $$3.4. $$What is the probability that the rate exceeds $$42$$?

Accepted Answer

0.1151

Question 8

Find the $$z-$$score for which the area under the standard normal curve to its left is $$0.04.$$

Accepted Answer

$$z = -1.75$$

Question 9

Determine the two $$z-$$scores that separate the middle $$87.4\% of $$the distribution from the area in the tails of the standard normal distribution.

Accepted Answer

$$z = \pm 1.51$$

Question 10

Assume $$X \sim N(50, 12^2$). $$Find the 24th percentile for $$X.$$

Accepted Answer

$$X = 50 + (-0.71) 	imes 12 = 41.48$$

Question 11

A physical fitness association wants to recognize the fastest 15% of boys in a mile run. If the mean time is $$440$$ seconds and the standard deviation is $$50$$ seconds, what time must a boy beat to earn a certificate?

Accepted Answer

$$440 + (-1.04) 	imes 50 = 387.96$$ seconds

Question 12

A sample of 700 non-fatal accidents showed that 143 involved faulty equipment. What is the point estimate for $$p$$, the population proportion?

Accepted Answer

$$\hat{p} = \frac{143}{700} = 0.204$$

Question 13

When 440 junior college students were surveyed, 200 said they have a passport. Construct a 95% confidence interval for the proportion of students with a passport. Which of the following is correct (rounded to three decimals)?

Accepted Answer

$$0.409 \u003c p \u003c 0.500$$

Question 14

A poll of 546 teenagers found that 37% want more discussion about school. Construct a 95% confidence interval for the proportion of all teenagers who want more family discussions about school.

Accepted Answer

$$0.329 \u003c p \u003c 0.411$$

Question 15

How large a sample is needed to be 95% confident that the sample proportion of left-handed scientists will not differ from the true proportion by more than 6%, given $$p = 0.09$$?

Accepted Answer

$$n = \left( \frac{1.96^2$ 	imes 0.09 	imes 0.91}{0.06^2$} ight) = 87$$

Question 16

Find the critical $$t-$$value for a 95% confidence interval with $$n = 16.$$

Accepted Answer

$$t_{0.025, 15} = 2.131$$

Question 17

A sample of 25 part-time workers had mean annual earnings of $$13,250$$ with a standard deviation of $$2,677. $$Construct a 95% confidence interval for the population mean (rounded to the nearest dollar).

Accepted Answer

$$13,250 \pm 1,048$$

Question 18

A doctor wants to estimate the mean birth weight of infants within 2.5 ounces with 90% confidence. If $$\sigma = 5.5$$ ounces, what sample size is needed?

Accepted Answer

$$n = \left( \frac{1.645 	imes 5.5}{2.5} \$$right)^2$$ = 13$$

Question 19

Which of the following best describes the sampling distribution of the sample mean for a population with mean $$\mu = 2.2$$ and standard deviation $$\sigma = 6$$, based on a sample of $$n = 100$$?

Accepted Answer

Approximately normal with mean $$2.2$$ and standard deviation $$0.6$$

Study Guide for Normal Distribution and Sampling in Statistics

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