Question
The number of books that high school students in the United States read in a year is normally distributed with a mean of 25 books and a standard deviation of 2.5 books. Complete the input for the spreadsheet program to calculate the probability that a high school student chosen at random will read more than 28 books in a year. Then identify this probability. Round the answers to the nearest tenth.(2 points)
A B
1 Mean
2 Standard Deviation
3 x
4 Cumulative TRUE
5
=NormDist (B3, B1, B2, B4)
%
The probability that a high school student chosen at random will read more than 28 books in a year is
a
%.
A B
1 Mean
2 Standard Deviation
3 x
4 Cumulative TRUE
5
=NormDist (B3, B1, B2, B4)
%
The probability that a high school student chosen at random will read more than 28 books in a year is
a
%.
Answers
GPT-4o mini
To calculate the probability that a high school student chosen at random will read more than 28 books in a year, you can follow these steps within your spreadsheet program:
1. Fill in the values as follows:
- B1 (Mean): 25
- B2 (Standard Deviation): 2.5
- B3 (x): 28
- B4 (Cumulative): TRUE (as it’s set for cumulative distribution)
2. The spreadsheet function you would use is `=NORM.DIST(B3, B1, B2, TRUE)`. This will give you the cumulative probability up to 28 books.
3. To find the probability of reading more than 28 books, you need to subtract the cumulative probability from 1:
\[
\text{Probability} = 1 - NORM.DIST(B3, B1, B2, TRUE)
\]
4. Applying this in your spreadsheet (let’s say you put the cumulative result in cell B5):
- In cell B6, you would input:
\[
=1 - B5
\]
### Calculation:
Using the normal distribution formula:
1. Calculate the cumulative probability for reading 28 books:
\[
Z = \frac{(X - \text{Mean})}{\text{Standard Deviation}} = \frac{(28 - 25)}{2.5} = \frac{3}{2.5} = 1.2
\]
2. Now, using a Z-score table or calculator:
\[
P(Z < 1.2) \approx 0.8849
\]
3. Now, to find the probability of reading more than 28 books:
\[
P(X > 28) = 1 - P(Z < 1.2) \approx 1 - 0.8849 = 0.1151
\]
4. Converting that to a percentage:
\[
0.1151 \times 100 = 11.51\%
\]
Rounded to the nearest tenth:
\[
\approx 11.5\%
\]
### Final answer:
The probability that a high school student chosen at random will read more than 28 books in a year is approximately **11.5%**.
1. Fill in the values as follows:
- B1 (Mean): 25
- B2 (Standard Deviation): 2.5
- B3 (x): 28
- B4 (Cumulative): TRUE (as it’s set for cumulative distribution)
2. The spreadsheet function you would use is `=NORM.DIST(B3, B1, B2, TRUE)`. This will give you the cumulative probability up to 28 books.
3. To find the probability of reading more than 28 books, you need to subtract the cumulative probability from 1:
\[
\text{Probability} = 1 - NORM.DIST(B3, B1, B2, TRUE)
\]
4. Applying this in your spreadsheet (let’s say you put the cumulative result in cell B5):
- In cell B6, you would input:
\[
=1 - B5
\]
### Calculation:
Using the normal distribution formula:
1. Calculate the cumulative probability for reading 28 books:
\[
Z = \frac{(X - \text{Mean})}{\text{Standard Deviation}} = \frac{(28 - 25)}{2.5} = \frac{3}{2.5} = 1.2
\]
2. Now, using a Z-score table or calculator:
\[
P(Z < 1.2) \approx 0.8849
\]
3. Now, to find the probability of reading more than 28 books:
\[
P(X > 28) = 1 - P(Z < 1.2) \approx 1 - 0.8849 = 0.1151
\]
4. Converting that to a percentage:
\[
0.1151 \times 100 = 11.51\%
\]
Rounded to the nearest tenth:
\[
\approx 11.5\%
\]
### Final answer:
The probability that a high school student chosen at random will read more than 28 books in a year is approximately **11.5%**.