Question

1. The reading speed of second grade students is approximately normal with a mean of 86 words per minute and a standard deviation of 12 words per minute.

a) What is the probability that a randomly selected student will read more than 95 words per minute?

b) What is the probability that a random sample of 12 second grade students results in a mean reading rate of more than 98 words per minute?

c) There is a 5% chance that the mean reading speed of a random sample of 20 second grade students will exceed what value?

Answer 1

Let X denote the reading speed of second grade students. i.e., X denotes the number of words read by a student in a minute.

It is given that X is approximately normal with a mean of 86 words per minute and a standard deviation of 12 words per minute.

i.e.,

(a) What is the probability that a randomly selected student will read more than 95 words per minute?

i.e., we have to find P(X > 95).

, {standardizing X using the formula }

, { On standardising X, we get the standard normal distribution, Z}

, {From standard normal tables, }

Therefore, the probability that a randomly selected student will read more than 95 words per minute = 0.2266.

(b) What is the probability that a random sample of 12 second grade students results in a mean reading rate of more than 98 words per minute?

Let denote the mean reading rate of a sample of n students.

We know the result that

if , then for any sample size n, the sampling distribution of is also normal, with mean µ and variance , i.e., ----------result (1)

Here, we have to find the probability that a random sample of size12 has a mean reading rate of more than 98 words per minute. i.e., we have to calculate , when .

Using result (1), we have , { since }

i.e., .

Now, consider , {standardizing }

, { On standardising , we get the standard normal distribution, Z}

, {From standard normal tables, }

Therefore, the probability that a random sample of 12 second grade students results in a mean reading rate of more than 98 words per minute = 0.0003

(c) There is a 5% chance that the mean reading speed of a random sample of 20 second grade students will exceed what value?

Here we have to find a constant c such that , where denotes the mean reading speed of a sample of 20 second grade students.

Using result (1), we have , { since }.

i.e.,

Consider

, {standardizing }

, where , { On standardising , we get the standard normal distribution, Z}

, {From standard normal tables, }

Therefore,

, {since c is the representation of number of words read in a minute, it has to be a whole number value. Therefore rounding to the nearest whole number}

Therefore, there is approximately a 5% chance that the mean reading speed of a random sample of 20 second grade students will exceed 90 words per minute.