# AP Statistics : Normal Distribution

## Example Questions

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### 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.

0.0822

.0586

0.3220

0.0768

0.9000

.0586

Explanation:

### 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

Alex = .945

Noah = .933

Alex = .923

Noah = .911

Alex = .901

Noah = .926

Alex = .778

Noah = .723

Alex = .855

Noah = .844

Alex = .945

Noah = .933

Explanation:

Alex -

on the z-table

Noah -

on the z-table

When

and

Find

.

.76

.68

.61

.72

.81

.72

Explanation:

### Example Question #1 : 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?

Explanation:

### Example Question #1 : 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  percentile of the masses of all the tomatoes?

Explanation:

The Z score for a normal distribution at the  percentile is  So  , which can be found on the normal distribution table. The mass of tomatoes in the  percentile of all tomatoes  is  standard deviations below the mean, so the mass is .

### Example Question #1 : Normal Distribution

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,

.

### Example Question #1 : Normal Distribution

Find

.

Explanation:

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

This gives us a p-value of,

.

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

This gives us a p-value of,

.

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,

.

### Example Question #1 : How To Use Tables Of 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.)

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:

where,

Plugging into the equation we get:

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.

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

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.

### Example Question #2 : 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?

Mean

Both

Variance

Standard Deviation