Question

The distribution of the amount of money spent by students for textbooks in a semester is approximately normal in shape with a mean of $235 and a standard deviation of $20. According to the standard deviation rule, almost 2.5% of the students spent more than ___________ on textbooks in a semester.A. $195B. $215C. $235D. $275E. $295

Answer 1

Answer:

D. $275

Step-by-step explanation:

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$, the zscore of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

$\mu = 235, \sigma = 20$

According to the standard deviation rule, almost 2.5% of the students spent more than ... on textbooks in a semester:

This is the value of X when Z has a pvalue of 1-0.025 = 0.975. So it is X when Z = 1.96[/tex]

So

$Z = \frac{X - \mu}{\sigma}$

$1.96 = \frac{X - 235}{20}$

$X - 235 = 20*1.96$

$X = 274.2$

So the correct answer is:

D. $275