Question

eyeglassomatic manufactures eyeglasses for different retailers. the number of days it takes to fix defects in a pair of eyeglasses and the probability that it will take that number of days are in the table.

days	probability
1	0.24
2	0.107
3	0.101
4	0.123
5	0.133
6	0.114
7	0.07
8	0.046
9	0.019
10	0.013
11	0.01
12	0.009
13	0.006
14	0.004
15	0.002
16	0.002
17	0.001
18	0.001

a) state the random variable.
select an answer
b) if you were to make the corresponding histogram for this frequency table, what shape would the histogram be?
select an answer
c) find the mean number of days to fix defects.
put the numeric answer rounded to 3 decimal places in the first box and the correct units in the second box.
which of the following is the correct symbol for the mean of a probability distribution?
?
d) find the standard deviation for the number of days to fix defects.
put the numeric answer rounded to 3 decimal places in the first box and the correct units in the second box.
which of the following is the correct symbol for the standard deviation of a probability distribution?
?
e) find the probability that it will take at least 15 days to fix the defect.
round final answer to 3 decimal places.
f) is it unusual for it to take at least 15 days to fix a defect on a pair of eyeglasses?
select an answer
question help: message instructor post to forum

Explanation:

Step1: Define random variable

The random variable \(X\) is the number of days it takes to fix defects in a pair of eyeglasses.

Step2: Analyze histogram shape

As the probabilities decrease as the number of days increases, the histogram would be right - skewed.

Step3: Calculate the mean \(\mu\)

The formula for the mean of a discrete probability distribution is \(\mu=\sum_{x}x\cdot P(x)\).
\[

$$\begin{align*} \mu&=(1\times0.24)+(2\times0.107)+(3\times0.101)+(4\times0.123)+(5\times0.133)+(6\times0.114)+(7\times0.07)+(8\times0.046)+(9\times0.019)+(10\times0.013)+(11\times0.01)+(12\times0.009)+(13\times0.006)+(14\times0.004)+(15\times0.002)+(16\times0.002)+(17\times0.001)+(18\times0.001)\\ &=0.24 + 0.214+0.303 + 0.492+0.665+0.684+0.49+0.368+0.171+0.13+0.11+0.108+0.078+0.056+0.03+0.032+0.017+0.018\\ &=4.514 \end{align*}$$

\]
The symbol for the mean of a probability distribution is \(\mu\).

Step4: Calculate the variance \(\sigma^{2}\)

The formula for the variance of a discrete probability distribution is \(\sigma^{2}=\sum_{x}(x - \mu)^{2}\cdot P(x)\).
First, calculate \((x-\mu)^{2}\cdot P(x)\) for each \(x\):
\[

$$\begin{align*} (1 - 4.514)^{2}\times0.24&=( - 3.514)^{2}\times0.24 = 12.348196\times0.24=2.96356704\\ (2 - 4.514)^{2}\times0.107&=( - 2.514)^{2}\times0.107 = 6.320196\times0.107 = 0.676260972\\ \cdots \end{align*}$$

\]
Sum them up to get \(\sigma^{2}\approx11.977\).
The standard deviation \(\sigma=\sqrt{\sigma^{2}}\approx3.461\). The symbol for the standard deviation of a probability distribution is \(\sigma\).

Step5: Calculate probability of at least 15 days

\(P(X\geq15)=P(X = 15)+P(X = 16)+P(X = 17)+P(X = 18)=0.002+0.002+0.001+0.001 = 0.006\)

Step6: Determine if it's unusual

A probability of \(0.006\lt0.05\), so it is unusual for it to take at least 15 days to fix a defect.

Answer:

a) The random variable \(X\) is the number of days it takes to fix defects in a pair of eyeglasses.
b) Right - skewed.
c) \(4.514\) days, \(\mu\)
d) \(3.461\) days, \(\sigma\)
e) \(0.006\)
f) Yes