Question

Use the normal distribution to approximate the desired probability. A certain question on a test is answered correctly by 25.0 percent of the respondents. Estimate the probability that among the next 140 responses there will be at most 41 correct answers.

Answer 1

Answer:

The probability that there will be at most 41 correct answers is 0.8790

Step-by-step explanation:

We can aproximate the probability by using a binomial distribution where:

p: A question on a test is answered correctly

n: number of responses

So, the mean of the distribution is given by:

$\mu= n\times p= 140 \times 0.25=35$

and the standar deviation is given by:

$\sigma=\sqrt{n\times p\times q} =\sqrt{n\times p\times (1-p)} =\sqrt{140\times 0.25\times 0.75} =5.123$

The normalized variable for 41 correct answers is:

$z=\frac{x-\mu}{\sigma}=\frac{41-35}{5.123} =1.17$

Hence, the probability that there will be at most 41 correct answers is:

$P(x$