Question

Assume that females have pulse rates that are normally distributed with a mean of mu equals 76.0 beats per minute and a standard deviation of sigma equals 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 between 70 beats per minute and 82 beats per minute. The probability is nothing. ​(Round to four decimal places as​ needed.) b. If 16 adult females are randomly​ selected, find the probability that they have pulse rates with a mean between 70 beats per minute and 82 beats per minute. The probability is nothing. ​(Round to four decimal places as​ needed.) 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 distribution is of​ individuals, not sample​ means, the distribution is a normal distribution for any sample size. D. Since the original population has a normal​ distribution, the distribution of sample means is a normal distribution for any sample size.

Answer 1

Solution :

Given that ,

mean = = 76.0

standard deviation = = 12.5

a) P(70 < x < 82) = P[(70 - 76.0)/12.5 ) < (x - ) /  < (82 - 76.0) / 12.5) ]

= P(-0.48 < z < 0.48)

= P(z < 0.48) - P(z < -0.48)

Using z table,

= 0.6844 - 0.3156

= 0.3688

b) n = 16

= = 76.0

= / n = 12.5 / 16 = 3.125

P(70 < < 82)  

= P[(70 - 76.0) /3.125 < ( - ) / < (82 - 76.0) / 3.125)]

= P(-1.92 < Z < 1.92)

= P(Z < 1.92) - P(Z < -1.92)

Using z table,  

= 0.9726 - 0.0274

= 0.9452

c) D. Since the original population has a normal​ distribution, the distribution of sample means is a normal distribution for any sample size.