Question

A psychologist designed a new aptitude exam to measure logical and analytical thinking abilities. The time allowed for the exam is 60 minutes, and the exam is made up of 35 multiple choice questions. The psychologist expects that an examinee will spend an average of 1.66 minutes answering each question, with a standard deviation of 0.73 minutes. What proportion of examinees will complete the exam on time? Carry your intermediate computations to at least four decimal places. Report your result to at least three decimal places

Answer 1

Answer:

52.79% of examinees will complete the exam on time.

Step-by-step explanation:

Problems of normally distributed samples can be 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:

There are 35 questions. The psychologist expects that an examinee will spend an average of 1.66 minutes answering each question, with a standard deviation of 0.73 minutes. This means that the mean and the standard deviation for the time to complete the test is $\mu = 35*1.66 = 58.1, \sigma = 35*0.73 = 25.55$.

What proportion of examinees will complete the exam on time?

The time allowed for the exam is 60 minutes. So this proportion is the pvalue of Z when $X = 60$. So:

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

$Z = \frac{60 - 58.1}{25.55}$

$Z = 0.07$

$Z = 0.07$ has a pvalue of 0.5279.

This means that 52.79% of examinees will complete the exam on time.