Answer:
a) This is the p-value of Z when X = A subtracted by the p-value of Z when X = B.
b) P-value of Z when X = 76 subtracted by the p-value of X = 68.
c) Because the underlying distribution(pulse rates of females) is normal.
Step-by-step explanation:
Normal Probability Distribution:
Problems of normal distributions can be solved using the z-score formula.
In a set with mean and standard deviation
, the z-score of a measure X is given by:
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.
Central Limit Theorem
The Central Limit Theorem establishes that, for a normally distributed random variable X, with mean and standard deviation
, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
and standard deviation
.
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
In this question:
Mean , standard deviation
.
standard deviation of beats per minute.
a. If 1 adult female is randomly selected, find the probability that her pulse rate is between A beats per minute and B beats per minute.
This is the p-value of Z when X = A subtracted by the p-value of Z when X = B.
b. If 4 adult females are selected, find the probability that they have pulse rates with a mean between 68 beats per minute and 76 beats per minute.
Sample of 4 means that we have
The formula for the z-score is:
This probability is the p-value of Z when X = 76 subtracted by the p-value of X = 68.
c. Why can the normal distribution be used in heartbeat even the sample side does not exceed 30?
Because the underlying distribution(pulse rates of females) is normal.