Question

Suppose that the distribution of scores on an exam is mound shaped and approximately symmetric. The exam scores have a mean of 110 and the 16th percentile is 85. (Use the Empirical Rule.)

(a)

What is the 84th percentile?

(b)

What is the approximate value of the standard deviation of exam scores?

(c)

What is the z-score for an exam score of 90?

(d)

What percentile corresponds to an exam score of 160?

%

(e)

Do you think there were many scores below 35? Explain.

Since a score of 35 is  ---Select--- one standard deviation two standard deviations  three standard deviations  below the mean, that corresponds to a percentile of  %. Therefore, there were  ---Select--- many few scores below 35.

Answer 1

Before, I answer your question let me define what empirical rule means.

The empirical rule states that for a normal distribution, nearly all the data will fall within three standard deviation from the mean.

• 68% of data falls within one standard deviation from the mean.
• 95% of data falls within two standard deviation from the mean.
• 99.7% of data falls within three standard deviation from the mean.

The empirical rule is also known as 68-95-99.7 rule or three sigma rule.

Please refer the above image for %age references in question.

Note: Entire area under the curve will be equal to 1. As the distribution is symmetric equal percentage of distribution lies on both the ends of distribution.

Now, as 68% of the data lies under one standard deviation from the mean, this implies that remaining 32% lies outside the one standard deviation from the mean and 16% lies below μ- and 16% lies above μ+.

Given μ=110 and P16=85.

First let us solve part(b) as it will help in part (a).

(b) 16th percentile implies that 16% of the data lies below this number.

which implies,

μ-= 16th percentile =P16

μ-=85

110-=85

=110-85

Standard deviation =25

(a) Now, μ=110 and =25

84th percentile means 84% data lies below it and 16% lies above it.

From Empirical rule, 84% percentile mean μ+.

Therefore, P84=μ+=110+25 = 135

(c) As, μ=110 and =25

Z score at x=90

z= (x-μ)/ = (90-110)/25 = -20/25 = -0.8

(d) Let score 160 corresponds to nth percentile.

According to empirical rule, 95% percentile will be given by μ+2*

P95= μ+2* = 110+ 2*25 = 110+50 = 160

Therefore score 160 corresponds to 95th percentile.

(e) We need to find how many scores are below 35.

By 3 rule,

2.5% of the data lie below= μ-2*= 110-2*25 = 110-50 =65

0.15% of the data lie below =μ-3*= 110-3*25 = 110-75 =35

Hence, 0.15% of the scores lies below 35.