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AP Statistics : Normal Distribution

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Example Questions

Example Question #1 : How To Use Tables Of Normal Distribution

Find the area under the standard normal curve between Z=1.5 and Z=2.4.

Possible Answers:

0.3220

0.0822

0.9000

0.0768

.0586

Correct answer:

.0586

Explanation:

$Z_{2.4}=.9918$

$Z_{1.5}=.9332$

.9918-.9332=.0586

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Example Question #1 : Normal Distribution

Alex took a test in physics and scored a 35. The class average was 27 and the standard deviation was 5.

Noah took a chemistry test and scored an 82. The class average was 70 and the standard deviation was 8.

Show that Alex had the better performance by calculating -

1) Alex's standard normal percentile and

2) Noah's standard normal percentile

Possible Answers:

Alex = .901

Noah = .926

Alex = .923

Noah = .911

Alex = .855

Noah = .844

Alex = .945

Noah = .933

Alex = .778

Noah = .723

Correct answer:

Alex = .945

Noah = .933

Explanation:

Alex -

$Z=\left ( 35-27\right )/5 = 1.6 = .945$ on the z-table

Noah -

$Z=\left ( 82-70\right )/8=1.5=.933$ on the z-table

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Example Question #2 : How To Use Tables Of Normal Distribution

When

$\mu=19$

and

$\sigma=6$

Find

$P\left ( 12< x< 25\right )$ .

Possible Answers:

.68

.76

.81

.61

.72

Correct answer:

.72

Explanation:

$Z_{12}=\left ( 12-19\right )\div6=-1.17$

$p_{-1.17}=.121$

$Z_{25}=\left ( 25-19\right )\div6=1$

$p_{1}=.8413$

$P\left ( 12< x< 25\right )=.8413 - .121 = .7203$

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Example Question #2 : Normal Distribution

Arrivals to a bed and breakfast follow a Poisson process. The expected number of arrivals each week is 4. What is the probability that there are exactly 3 arrivals over the course of one week?

Possible Answers:

$19.5\%$

$41\%$

$12.4\%$

$9.23\%$

$63.5\%$

Correct answer:

$19.5\%$

Explanation:

$P(X=3)=\frac{(\lambda t)^n}{n!}e^{-\lambda t}=\frac{(4\cdot 1)^3}{3!}e^{-4\cdot 1}=19.5\%$

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Example Question #1 : How To Use Tables Of Normal Distribution

The masses of tomatoes are normally distributed with a mean of grams and a standard deviation of grams. What mass of tomatoes would be the 25th percentile of the masses of all the tomatoes?

Possible Answers:

42.3 g

46.2g

42.0g

43.3g

44.2g

Correct answer:

42.3 g

Explanation:

The Z score for a normal distribution at the 25th percentile is z = -0.675. So $P(z < -0.675) \approx 0.25$ , which can be found on the normal distribution table. The mass of tomatoes in the 25th percentile of all tomatoes is 0.675 standard deviations below the mean, so the mass is $45g - 0.675 \ast 4g \approx 42.3g$ .

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Example Question #2 : Normal Distribution

Find

P(0.50 < z < 1.23) .

Possible Answers:

0.1

0.3123

0.4512

0.1992

Correct answer:

0.1992

Explanation:

First, we use our normal distribution table to find a p-value for a z-score greater than 0.50.

Our table tells us the probability is approximately,

P(z>0.50)=0.6915 .

Next we use our normal distribution table to find a p-value for a z-score greater than 1.23.

Our table tells us the probability is approximately,

P(z<1.23)=0.8907 .

We then subtract the probability of z being greater than 0.50 from the probability of z being less than 1.23 to give us our answer of,

0.8907-0.6915=0.1992 .

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Example Question #3 : Normal Distribution

Find

P(-1.22 < z < 1.59) .

Possible Answers:

0.8329

0.342

0.965

0.721

Correct answer:

0.8329

Explanation:

First, we use the table to look up a p-value for z > -1.22.

This gives us a p-value of,

P(z>-1.22)=0.1112 .

Next, we use the table to look up a p-value for z > 1.59.

This gives us a p-value of,

P(z<1.59)=0.9441 .

Finally we subtract the probability of z being greater than -1.22 from the probability of z being less than 1.59 to arrive at our answer of,

0.9441-0.1112=0.8329 .

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Example Question #3 : Normal Distribution

Gabbie earned a score of 940 on a national achievement test. The mean test score was 850 with a sample standard deviation of 100. What proportion of students had a higher score than Gabbie? (Assume that test scores are normally distributed.)

Possible Answers:

0.10

0.1841

0.82

0.5012

Correct answer:

0.1841

Explanation:

When we get this type of problem, first we need to calculate a z-score that we can use in our table.

To do that, we use our z-score formula:

$Z = \frac{(X - \mu)}{\sigma}$

where,

$\\X=score=940 \\ \mu=mean=850 \\ \sigma = sample\ standard\ deviation=100$

Plugging into the equation we get:

$Z=\frac{(940 - 850)}{100}=\frac{90}{100}=0.90$

We then use our table to look up a p-value for z > 0.9. Since we want to calculate the probability of students who earned a higher score than Gabbie we need to subtract the P(z<0.9) to get our answer.

$\\P(z>0.90)=1-P(z<0.90) \\P(z>0.90) =1-0.8159 \\P(z>0.90)=0.1841$

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Example Question #2 : Normal Distribution

Which parameters define the normal distribution?

Possible Answers:

$\lambda, N$

$\mu, \bar{x}$

$\mu, N$

$\mu, \sigma$

Correct answer:

$\mu, \sigma$

Explanation:

The two main parameters of the normal distribution are $\mu$ and $\sigma$ . $\mu$ is a location parameter which determines the location of the peak of the normal distribution on the real number line. $\sigma$ is a scale parameter which determines the concentration of the density around the mean. Larger $\sigma$ 's lead the normal to spread out more than smaller $\sigma$ 's.

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Example Question #1 : How To Identify Characteristics Of A Normal Distribution

All normal distributions can be described by two parameters: the mean and the variance. Which parameter determines the location of the distribution on the real number line?

Possible Answers:

Standard Deviation

Both

Mean

Variance

Correct answer:

Mean

Explanation:

The mean determines where the normal distribution lies on the real number line, while the variance determines the spread of the distribution.

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AP Statistics : Normal Distribution

Study concepts, example questions & explanations for AP Statistics

All AP Statistics Resources

Example Questions

Example Question #1 : How To Use Tables Of Normal Distribution

Example Question #1 : Normal Distribution

Example Question #2 : How To Use Tables Of Normal Distribution

Example Question #2 : Normal Distribution

Example Question #1 : How To Use Tables Of Normal Distribution

Example Question #2 : Normal Distribution

Example Question #3 : Normal Distribution

Example Question #3 : Normal Distribution

Example Question #2 : Normal Distribution

Example Question #1 : How To Identify Characteristics Of A Normal Distribution

All AP Statistics Resources

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