Question

Eyeglassomatic manufactures eyeglasses for different retailers. In March 2010, they tested to see how many defective lenses they made, and there were 16.9% defective lenses due to scratches. Suppose Eyeglassomatic examined twenty eyeglasses.

a.) State the random variable.
b.) Write the probability distribution.
c.) Draw a histogram.
d.) Describe the shape of the histogram.
e.) Find the mean.
f.) Find the variance.
g.) Find the standard deviation.

Answer 1

a) The random variable is the number of defective lenses in a sample of 20 eyeglasses. b) The probability distribution can be found using the binomial probability formula. c) The histogram can be drawn by plotting the values of the random variable and their probabilities.

Step-by-step explanation:

a) The random variable in this case is the number of defective lenses in a sample of 20 eyeglasses.

b) To write the probability distribution, we need to find the probabilities for each possible value of the random variable. The probability of having a defective lens is given as 16.9%. To find the probability of having k defective lenses in a sample of 20 eyeglasses, we use the binomial probability formula: P(X=k) = (n choose k) * p^k * (1-p)^(n-k), where n is the number of trials (20), k is the number of successes (defective lenses), and p is the probability of success (16.9%). Substitute the values into the formula to find the probabilities for k=0, 1, 2, ..., 20.

c) To draw a histogram, we plot the values of the random variable on the x-axis and the corresponding probabilities on the y-axis.

d) The shape of the histogram is likely to be skewed to the right, since the probability of having more defective lenses is relatively small compared to the probability of having fewer defective lenses.

e) The mean is calculated by multiplying each possible value of the random variable by its corresponding probability, and then summing them up. In this case, the mean is given by: mean = (0 * P(X=0)) + (1 * P(X=1)) + ... + (20 * P(X=20)).

f) The variance is calculated by taking the squared difference between each possible value of the random variable and the mean, multiplying it by the corresponding probability, and then summing them up. In this case, the variance is given by: variance = (0 - mean)^2 * P(X=0) + (1 - mean)^2 * P(X=1) + ... + (20 - mean)^2 * P(X=20).

g) The standard deviation is the square root of the variance. Calculate the standard deviation by taking the square root of the variance obtained in part f).