Question

Assume that females have pulse rates that are normally distributed with a mean of 75.0 beats per minute and a standard deviation of 12.5 beats per minute. Complete parts (a) through (b) below. af 16 adult females are randomly selected, find the probability that they have pulse rates with a sample mean less than 78 beats per minute The probability is (Round to four decimal places as needed.) b. Why can the normal distribution be used in part (a), even though the sample size does not exceed 307 O A Since the mean pulse rate exceeds 30, the distribution of sample means is a normal distribution for any sample size 3. Since the distribution is of individuals, not sample means, the distribution is a normal distribution for any sample size O C. Since the distribution is of sample means, not individuals, the distribution is a normal distribution for any sample size OD. Since the original population has a normal distribution, the distribution of sample means is a normal distribution for any sample size

Answer 1

Here we have given that

Xi: Pulse rates of females in beats per minute

$\mu$ = Populaiton mean = 75 beats per minute

$\sigma =$ Population Standard deviation = 12.5 beats per minute

(A)

Now we want to calculate the probability that adult women have pulse rates with a sample mean less than 78 beats per minute?

i.e $P( \bar x < 78)$ =?

$P( \bar x < 78)$ = $P (z < \frac{\bar x- \mu } { \frac{\sigma}{\sqrt(n)}})$

= $P (z < \frac{78-75 } { \frac{12.5}{\sqrt(16)}})$

=$P (z < \frac{3 } {3.125})$

=

=0.8315 Using standard normal z table see value corresponding z score 0.96

The probability that adult women have pulse rates with a sample mean less than 78 beats per minute is 0.8315

i.e. $P( \bar x < 78)$ = 0.8315

(B)

Here, we have given that

Assume that the females have pulse rates that are normally distributed with population parameters which refers that the original population distribution is normal and then we took the 16 females sample randomly from the population.

Hence, here option D is correct

Since the original population has a normal distribution. the distribution of sample means is the normal distribution for any sample size.