Question

The quantitative data set under consideration has roughly a​ bell-shaped distribution. Apply the empirical rule to answer the following question. A quantitative data set of size 100 has mean 50 and standard deviation 4. Approximately how many observations lie between 38 and 62​?

Answer 1

Approximately 95 of the 100 observations lie between 38 and 62.

Step-by-step explanation:

The empirical rule states that approximately 95% of the data in a bell-shaped distribution lies within two standard deviations of the mean. In this case, the data set has a mean of 50 and a standard deviation of 4. To find the number of observations between 38 and 62, we need to calculate the z-scores for these values.

The z-score for 38 is (38 - 50) / 4 = -3, and the z-score for 62 is (62 - 50) / 4 = 3. Using the empirical rule, we know that approximately 95% of the observations lie within two standard deviations of the mean, which corresponds to z-scores of -2 and 2.

So, the number of observations between 38 and 62 can be estimated as the proportion of the total data set that lies between z-scores of -2 and 2, which is approximately 95%. Therefore, approximately 95 of the 100 observations lie between 38 and 62.

Answer 2

Approximately 100 lie between 38 and 62.

Step-by-step explanation:

The Empirical Rule states that, for a normally distributed(bell-shaped) random variable:

68% of the measures are within 1 standard deviation of the mean.

95% of the measures are within 2 standard deviation of the mean.

99.7% of the measures are within 3 standard deviations of the mean.

In this problem, we have that:

100 observations.

Mean = 50

Standard deviation = 4

Approximately how many observations lie between 38 and 62​?

38 = 50 - 4*3

38 is three standard deviations below the mean

62 = 50 + 4*3

62 is three standard deviations above the mean

By the Empirical Rule, 99.7% of the measures are within 3 standard deviations of the mean.

There are 100 observations.

0.997*100 = 99.7

Approximately 100 lie between 38 and 62.