Final answer:
To find the probability of an adult having a pulse rate over 77 bpm, calculate the Z-score and refer to a Z-table or normal distribution calculator. Similarly, for the probability of a sample mean of 19 adults exceeding 77 bpm, calculate the Z-score using the standard error of the mean and consult the Z-table.
Step-by-step explanation:
Finding Probability in a Normal Distribution
To solve these problems, we will apply principles of the normal distribution using the provided mean and standard deviation.
a. Probability of an Adult Having a Pulse Rate Over 77 bpm:
The Z-score is calculated as follows:
Z = (X - μ) / σ
Where X is the value of interest (77 bpm), μ is the mean (72 bpm), and σ is the standard deviation (11 bpm).
Z = (77 - 72) / 11 = 0.4545
Using a Z-table or normal distribution calculator, we find the probability of a Z-score being above 0.4545. This value is equivalent to 1 minus the probability of Z being less than 0.4545.
b. Probability of a Random Sample of 19 Adults Having a Mean Over 77 bpm:
When considering a sample, we use the standard error of the mean (SEM), which is σ/√n, where n is the sample size.
SEM = 11 / √19 = 2.525
We calculate the Z-score using the SEM:
Z = (77 - 72) / 2.525 = 1.9802
Again, we'd look up the probability associated with this Z-score to find the probability of the mean of a sample of 19 adults being over 77 bpm.