Question

Due Sun 03/08/2020 11:59 pm The Acme Company manufactures widgets. The distribution of widget weights is bell-shaped. The widget weights have a mean of 37 ounces and a standard deviation of 8 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 and b) What percentage of the widget weights lie between 13 and 53 ounces? c) What percentage of the widget weights lie above 29? Get help: Video Points possible: 10 This is attempt 1 of 3. Submit

Answer 1

The empirical rule states that for a normal distribution:

68% of the data will fall within one standard deviation of the mean.

95% of the data will fall within two standard deviations of the mean.

99.7% of the data will fall within three standard deviations of the mean.

Q1:

µ = 37, σ = 8

a) (µ - 2*σ, µ + 2*σ)

(37 - 2*8, 37 + 2*8)

(21, 53)

95% of the widget weights lie between 21 and 53.

b)

z = (X-µ)/σ = (13-37)/8 = -3

z = (X-µ)/σ = (53-37)/8 = 2

Percentage of the widget weights between 13 and 53 = 99.7/2 + 95/2 = 97.35%

c)

z score = (X-µ)/σ = (29-37)/8 = -1

Percentage of widget weight lie above 29 = 68 + (100-68)/2 = 84%

---------------

Q2:

µ = 48, σ = 8

a)

(µ - 2*σ, µ + 2*σ)

(48 - 2*8, 48 + 2*8)

(32, 64)

95% of the widget weights lie between 32 and 64.

b)

z = (X-µ)/σ = (40-48)/8 = -1

z = (X-µ)/σ = (64-48)/8 = 2

Percentage of the widget weights between 40 and 64 = 68/2 + 95/2 = 81.5%

c)

z score = (X-µ)/σ = (72-37)/8 = 3

Percentage of widget weight lie above 29 = (100-99.7)/2 = 0.15%

----------

Q3:

Probability of win = 1/821 = 0.001218

Probability of lose, q = 1 - p = 1 - 0.001218 = 0.998782

Amount lost = -7

Amount gained = 1700

Expected value of the policy for insurance company = p*gain +q*lose

= 0.001218 * 1700 + 0.998782 * (-7)

= $ -4.92