Question

The Acme Company manufactures widgets. The distribution of widget weights is bell-shaped. The widget weights have a mean of 52 ounces and a standard deviation of 5 ounces. Use the Standard Deviation Rule, also known as the Empirical Rule.

Suggestion: sketch the distribution in order to answer these questions.

a) 95% of the widget weights lie between *blank* and *blank*

b) What percentage of the widget weights lie between 37 and 62 ounces?

c) What percentage of the widget weights lie above 47 ?

Answer 1

Let the random variable X denote the widget weights manufactured by the Acme Company. We are given, that the distribution of widget weights is bell-shaped, i.e symmetric and hence can be claimed to be normally distributed and

Mean of X () = 52 ounces and standard deviation of X () = 5 ounces

X ~ N (52,5)

By the empirical rule, 68% of data falls within the first standard deviation from the mean. 95% fall within two standard deviations. 99.7% fall within three standard deviations from the mean.

a) From the statement mentioned above,

95% of the widget weights would lie between

= (52 - 10 , 52 + 10)

= (42, 62)

Hence,  95% of the widget weights would lie between _42_ and _62_ ounces.

b) Percentage of the widget weights that lie between 37 and 62 ounces

= Pr(37 < X < 62)

By definition of standard normal variate

Hence, obtaining the probabilities would be much easier we determine them using a standard normal distribution, with mean zero and variance 1:

Required percentage

Since, the standard normal table gives only the left tail probabilities (area),

From Normal table,

= 0.97725 - 0.00135

= 0.9759

Percentage of the widget weights that lie between 37 and 62 ounces would be 97.59%

c) .Percentage of the widget weights lie above 47

= Pr(X > 47)

= P(Z > -1)

= 1 -

From normal table,

= 1 - 0.15866

= 0.84134

Percentage of the widget weights lie above 47 would be 84.134%.