Normal Distribution

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AP Statistics › Normal Distribution

Questions 1 - 10
1

Which parameters define the normal distribution?

Explanation

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

2

Which parameters define the normal distribution?

Explanation

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

3

Find

.

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,

.

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,

.

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,

.

4

Find

.

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,

.

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,

.

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,

.

5

In order to be considered a normal distribution, a data set (when graphed) must follow a bell-shaped symmetrical curve centered around the mean.

It must also adhere to the empirical rule that indicates the percentage of the data set that falls within (plus or minus) 1, 2 and 3 standard deviations of the mean.

In order to be a normal distribution, what percentage of the data set must fall within:

  1. 68

  2. 95

  3. 99.7

  1. 65

  2. 98

  3. 99.7

  1. 68

  2. 95

  3. 98

  1. 68

  2. 95

  3. 98.7

  1. 65

  2. 98

  3. 99.9

Explanation

  1. Percentile for Z=1 is .8413 - or - .1587 in one tail - or - .3174 in both tails -

1 - .3174=.6826

  1. Percentile for Z=2 is .9772 - or - .0228 in one tail - or - .0456 in both tails -

1 - .0456=.9544

  1. Percentile for Z=3 is .9987 - or - .0013 in one tail - or - .0026 in both tails -

1 - .0026=.9974

6

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

.0586

0.9000

0.3220

0.0768

0.0822

Explanation

7

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

Alex = .945

Noah = .933

Alex = .923

Noah = .911

Alex = .901

Noah = .926

Alex = .855

Noah = .844

Alex = .778

Noah = .723

Explanation

Alex -

on the z-table

Noah -

on the z-table

8

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

Alex = .945

Noah = .933

Alex = .923

Noah = .911

Alex = .901

Noah = .926

Alex = .855

Noah = .844

Alex = .778

Noah = .723

Explanation

Alex -

on the z-table

Noah -

on the z-table

9

In order to be considered a normal distribution, a data set (when graphed) must follow a bell-shaped symmetrical curve centered around the mean.

It must also adhere to the empirical rule that indicates the percentage of the data set that falls within (plus or minus) 1, 2 and 3 standard deviations of the mean.

In order to be a normal distribution, what percentage of the data set must fall within:

  1. 68

  2. 95

  3. 99.7

  1. 65

  2. 98

  3. 99.7

  1. 68

  2. 95

  3. 98

  1. 68

  2. 95

  3. 98.7

  1. 65

  2. 98

  3. 99.9

Explanation

  1. Percentile for Z=1 is .8413 - or - .1587 in one tail - or - .3174 in both tails -

1 - .3174=.6826

  1. Percentile for Z=2 is .9772 - or - .0228 in one tail - or - .0456 in both tails -

1 - .0456=.9544

  1. Percentile for Z=3 is .9987 - or - .0013 in one tail - or - .0026 in both tails -

1 - .0026=.9974

10

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

.0586

0.9000

0.3220

0.0768

0.0822

Explanation

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