Question

1. although companies would like consumers to believe that identity theft protection is an essential expense, in reality it is discretionary. a consumer organization compared the monthly costs of similar identity theft protection plans on the market and published the following list: $14.99 $12.75 $14.99 $14.99 $9.99 $25 $25 $10 $14.99 $10 $20 $14.99 $10 $25 $20 $12 $14.99 $25 $25 $20 $12.75 $10 $20 $10 $20 $14.99 $10 $25 $20 $12 $14.99 $25 $25 $20 $12.75 $10 $9.99

a. write the formula for the mean in sigma notation and use it to calculate the mean monthly plan price. round your answer to the nearest cent.

b. construct a frequency distribution for the data.

c. use the frequency distribution to determine the mean. round to the nearest cent.

d. use the frequency distribution to determine the median and the mode.

1. thalia purchased a microphone and stand set on march 11 of a leap year. the microphone has a 1 - year warranty. the stand has a 120 - day warranty.

a. when is the last day the microphone warranty will be in effect? when will the warranty on the microphone expire?

b. when is the last day the microphone stand warranty will be in effect? when will the warranty on the stand expire?

Explanation:

Step1: Recall mean formula

The formula for the mean $\bar{x}$ of a set of data $x_1,x_2,\cdots,x_n$ in sigma - notation is $\bar{x}=\frac{1}{n}\sum_{i = 1}^{n}x_i$, where $n$ is the number of data - points and $x_i$ represents each data - point.

Step2: Count number of data - points

Count the number of prices. There are $n = 20$ prices: $9.99,10,10,10,10,12,12.75,12.75,14.99,14.99,14.99,14.99,14.99,20,20,20,25,25,25,25$.

Step3: Calculate sum of data - points

$\sum_{i = 1}^{20}x_i=9.99+4\times10 + 12+2\times12.75+5\times14.99+3\times20+4\times25$
$=9.99 + 40+12 + 25.5+74.95+60+100$
$=9.99+40+12+25.5+74.95+60+100 = 322.44$.

Step4: Calculate the mean

$\bar{x}=\frac{1}{n}\sum_{i = 1}^{n}x_i=\frac{322.44}{20}=16.122\approx16.12$.

Step5: Construct frequency distribution

Price ($x$)	Frequency ($f$)
10	4
12	1
12.75	2
14.99	5
20	3
25	4

Step6: Calculate mean using frequency distribution

The formula for the mean using frequency distribution is $\bar{x}=\frac{\sum_{i = 1}^{k}f_ix_i}{\sum_{i = 1}^{k}f_i}$, where $k$ is the number of distinct values, $f_i$ is the frequency of the $i$ - th value, and $x_i$ is the $i$ - th value.
$\sum_{i = 1}^{k}f_ix_i=1\times9.99+4\times10 + 1\times12+2\times12.75+5\times14.99+3\times20+4\times25$
$=9.99+40 + 12+25.5+74.95+60+100=322.44$ and $\sum_{i = 1}^{k}f_i=1 + 4+1+2+5+3+4 = 20$.
$\bar{x}=\frac{322.44}{20}=16.122\approx16.12$.

Step7: Find median using frequency distribution

Since $n = 20$ (an even number), the median is the average of the 10th and 11th ordered data - points.
Counting frequencies: $1+4+1+2 = 8$ (first 8 data - points), and $1+4+1+2+5=13$ (first 13 data - points). So the 10th and 11th data - points are both $14.99$. The median $M=\frac{14.99 + 14.99}{2}=14.99$.
The mode is the value with the highest frequency. The mode is $14.99$ (frequency of 5).

For question 16:

Step1: Determine microphone warranty expiration

A leap year has 366 days. Thalia purchased on March 11. One - year later, the last day of the microphone warranty is March 11 of the next year.

Step2: Determine stand warranty expiration

March has 31 days. From March 11 to March 31, there are $31 - 11=20$ days. April has 30 days, May has 31 days, June has 30 days. $20+30+31+30 = 111$ days. So the stand warranty expires on July 1 (120 - 111 = 9 days into July).

Answer:

a. The formula for the mean is $\bar{x}=\frac{1}{n}\sum_{i = 1}^{n}x_i$, and the mean is $\$16.12$.
b.

Price ($x$)	Frequency ($f$)
10	4
12	1
12.75	2
14.99	5
20	3
25	4

c. The mean using frequency distribution is $\$16.12$.
d. The median is $\$14.99$ and the mode is $\$14.99$.

1. a. The last day the microphone warranty will be in effect is March 11 of the next year.

b. The last day the microphone stand warranty will be in effect is July 1.