Assume that females have pulse rates that are normally distributed with a mean of u equals 75.0 beats per minute and a standard deviation of
σ=12.5 beats per minute. Complete parts (a) through (c) below.
a. If 1 adult female is randomly selected, find the probability that her pulse rate is less than 82 beats per minute. The probability is ___.
b. If 25 adult females are randomly selected, find the probability that they have pulse rates with a mean less than 82 beats per minute.
The probability is ____.
c. Why can the normal distribution be used in part (b), even though the sample size does not exceed 30?
A. Since the distribution is of sample means, not individuals, the distribution is a normal distribution for any sample size.
B. Since the mean pulse rate exceeds 30, the distribution of sample means is a normal distribution for any sample size.
C. Since the original population has a normal distribution, the distribution of sample means is a normal distribution for any sample size.
D. Since the distribution is of individuals, not sample means, the distribution is a normal distribution for any sample size.
Solution :
_{} = / n = 12.5 / 25 = 2.5
(a)
P(x < 82) = P[(x - ) / < (82 - 75) / 12.5]
= P(z < 0.56)
= 0.7123
(b)
P( < 82) = P(( - _{} ) / _{} < (82 - 75) / 2.5)
= P(z < 2.8)
= 0.9974
C. Since the original population has a normal distribution, the distribution of sample means is a normal distribution for any sample size.
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