Question

# 3. Weights of quarters are normally distributed with a mean of 5.67 g and a standard...

3. Weights of quarters are normally distributed with a mean of 5.67 g and a standard deviation of 0.06 g. Some vending machines are designed so that you can adjust the weights of quarters that are accepted. If many counterfeit coins are found, you can narrow the range of acceptable weights with the effect that most counterfeit coins are rejected along with some legitimate quarters.

3. a)  If you adjust your vending machines to accept weights between 5.60 g and 5.74 g, what percentage of legal quarters are accepted?

3. b)  If you adjust vending machines to accept all legal quarters except those with weights in the top 2.5% and the bottom 2.5%, what are the limits of the weights that are accepted?

Solution :

Given that,

mean = = 5.67

standard deviation = = 0.06

a ) P (5.60 < x < 5.74 )

P ( 5.60 -5.67/ 0.06) < ( x -  / ) < ( 5.74 -5.67/ 0.06)

P ( - 0.07 / 0.06 < z < 0.07 / 0.06 )

P (-1.17< z < 1.17 )

P ( z < 1.17) - P ( z < -1.17 )

Using z table

= 0.8790 - 0.1210

= 0.7580

Probability = 0.7580

b )P( Z > z) =2.5%

P(Z > z) = 0.025

1 - P( Z < z) = 0.025

P(Z < z) = 1 - 0.025

P(Z < z) = 0.975

z = 1.96

Using z-score formula,

x = z * +

x = 1.96 * 0.06 + 5.67

x = 5.79

c ) P(Z < z) = 2.5%

P(Z < z) = 0.025

P(Z < - 0.61) = 0.025

z = - 1.96

Using z-score formula,

x = z * +

x = -1.96 * 0.06 + 5.67

x = 5.55

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