Question

A) The Acme Company manufactures widgets. The distribution of widget weights is bell-shaped. The widget weights have a mean of 57 ounces and a standard deviation of 8 ounces.

Use the Empirical Rule and a sketch of the normal distribution in order to answer these questions.

68% of the widget weights lie between _____?____ and ______?______

  What percentage of the widget weights lie between 41 and 65 ounces?

What percentage of the widget weights lie above 33 ?

B) The Acme Company manufactures widgets. The distribution of widget weights is bell-shaped. The widget weights have a mean of 39 ounces and a standard deviation of 5 ounces.

Use the Empirical Rule and a sketch of the normal distribution in order to answer these questions.

95% of the widget weights lie between ___?____ and ___?____

What percentage of the widget weights lie between 34 and 49 ounces?

What percentage of the widget weights lie above 24 ?

Answer 1

A)

By Empirical rule, 68% of the data lies within 1 standard deviations from the mean.

By Empirical rule, 95% of the data lies within 2 standard deviations from the mean.

By Empirical rule, 99% of the data lies within 3 standard deviations from the mean.

68% of the widget weights lie between _____57-8 = 49____ and ______57+8 = 65____

Z score for x=41 is (41 - 57)/8 = -2

Z score for x = 65 is (65 - 57)/8 = 1

Percentage of widget weights lie between 41 and 65 ounces = Percentage of data lie within -2 and +1 standard deviation = 0.95/2 + 0.68/2 = 0.815

Z score for x=33 is (33 - 57)/8 = -3

Percentage of widget weights lie above 33 = 1 - Percentage of widget weights lie below 33 = 1 - Percentage of widget weights lie below -3 standard deviations from mean = 1 - (1 - 0.997)/2 = 0.9985

B)

95% of the widget weights lie between ___39 - 2 * 5 = 29____ and ___39 + 2 * 5 = 49

Z score for x = 34 is (34 - 39)/5 = -1

Z score for x = 49 is (49 - 39)/5 = 2

percentage of the widget weights lie between 34 and 49 ounces = Percentage of data within -1 and 2 standatrd deviation from mean = 0.68/2 + 0.95/2 = 0.815

Z score for x = 24 is (24 - 39)/5 = -3

percentage of the widget weights lie above 24 = 1 - Percentage of widget weights lie below 24 = 1 - Percentage of widget weights lie below -3 standard deviations from mean = 1 - (1 - 0.997)/2 = 0.9985