1. The reading speed of second grade students is approximately normal, with a me
ID: 3361047 • Letter: 1
Question
1. The reading speed of second grade students is approximately normal, with a mean of 90 words per minute (wpm) and a standard deviation of 10 wpm.
(a) What is the probability 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 95 words per minute?
(c) What is the probability that a random sample of 24 second grade students results in a mean reading rate of more than 95 words per minute?
(d) What effect does increasing the sample size have on the probability? Provide an explanation for this result.
(e) A teacher instituted a new reading program at school. After 10 weeks in the program, it was found that the mean reading speed of a random sample of 20 second grade students was 92.8 wpm. What might you conclude based on this result?
(f) There is a 5% chance that the mean reading speed of a random sample of 20 second grade students will exceed what value?
Explanation / Answer
Solution:- given that mean = 90 min sd = 10 min
a) the probability a randomly selected student will read more than 95 words per minute:
P( X > 95) = P( Z > (95 - 90)/10 )
= P( Z > 0.5)
= 1 P(Z < 0.5)
= 1 0.6915
= 0.3085
b) the probability that a random sample of 12 second grade students results in a mean reading rate of more than 95 words per minute:
P( X > 95) = P( Z > (95 - 90)/(10/sqrt(12)) )
= P( Z > 1.7321)
= 0.0418
c) What is the probability that a random sample of 24 second grade students results in a mean reading rate of more than 95 words per minute?
P( X > 95) = P( Z > (95 - 90)/(10/sqrt(24)) )
= P( Z > 2.4495)
= 0.0071
d) increasing the sample size decreases the probability because standard deviation of sample mean decreases as n increase
e) the mean reading speed of a random sample of 20 second grade students was 92.8 wpm
P( X > 95) = P( Z > (95 - 90)/(10/sqrt(20)) )
= P( Z > 2.2361 )
= 0.0125
f) 5% chance that the mean reading speed of a random sample of 20 second grade students :
P( Z > (X - 90)/(10/sqrt(20)) ) = 0.95
= > (X - 90)/(10/sqrt(20) = 1.645
X - 90 = 1.645 * 2.2361
X = 3.6784 + 90
X = 93.6784
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