Question

The distribution of the amount of money spent on book purchases for a semester by college students has a mean of $280 and a standard deviation of $40. If the distribution is bell-shaped and symmetric, what proportion of students will spend between $200 and $280 this semester? Round your answer to four decimal places.

Answer 1

Approximately 34% of the college students will spend between $200 and $280 on book purchases for a semester according to the empirical rule for a bell-shaped and symmetric distribution.

Step-by-step explanation:

The distribution of the amount of money spent on book purchases for a semester by college students is described as bell-shaped and symmetric with a mean of $280 and a standard deviation of $40. To find the proportion of students that will spend between $200 and $280, we can use the empirical rule (68-95-99.7 rule) which indicates that approximately 68% of the data fall within one standard deviation from the mean. Since we are concerned with the range from $200 (which is 2 standard deviations below the mean) to $280 (the mean), we are effectively looking at half of this 68% range. Therefore, approximately 34% of the students are expected to spend between $200 and $280 on books for a semester.

To calculate the proportion, we take 68% of the range (which covers -1 to +1 standard deviation from the mean) and divide it by 2:

• 68% / 2 = 34%

Thus, the proportion of students spending between $200 and $280 is 0.34, which when rounded to four decimal places, gives us 0.3400.

Answer 2

Answer: 0.4772

Step-by-step explanation:

Given : The distribution is bell shaped , then the distribution must be normal distribution.

Mean :
`\mu=\$280`

Standard deviation :
`\sigma= \$40`

The formula to calculate the z-score :-

`z=(x-\mu)/(\sigma)`

For x = $200

`z=(200-280)/(40)=-2`

For x = $280

`z=(280-280)/(40)=0`

The p-value =
`P(-2<z<0)=P(z<0)-P(z<-2)`

`0.5-0.0227501=0.4772499\approx0.4772`

Hence, the proportion of students will spend between $200 and $280 this semester = 0.4772