Suppose that people's heights (in centimeters) are normally distributed, with a mean of 175 and a standard deviation of 6. We find the heights of 40 people.

(a) How many would you expect to be between 169 and 181 cm tall?


(b) How many would you expect to be taller than 170 cm?

This is one of the best pages for your kind of question

http://davidmlane.com/hyperstat/z_table.html

enter 175 and 6 in the "mean" and "sd" boxes.
click on the "betwee" button and enter 169 and 181 to get a range of .682689

so the number of people out of 40 to be between 169 and 181 would be
.682689(40) or appr. 27

b) click on the "above" button and enter 170, then repeat as above

Suppose that people's heights (in centimeters) are normally distributed, with a mean of 175 and a standard deviation of 6. We find the heights of 40 people.

(a) How many would you expect to be between 167 and 183 cm tall?
1

(b) How many would you expect to be taller than 170 cm?
2

To answer these questions, we need to use the properties of the normal distribution and the given mean and standard deviation.

(a) To find the number of people expected to be between 169 and 181 cm tall, we need to calculate the area under the normal distribution curve between these two heights. We can use the z-score formula to standardize the heights and then look up the corresponding probabilities from the standard normal distribution table.

First, we calculate the z-scores for the lower and upper height limits:
z1 = (169 - mean) / standard deviation = (169 - 175) / 6 = -1
z2 = (181 - mean) / standard deviation = (181 - 175) / 6 = 1

Next, we look up the corresponding probabilities from the standard normal distribution table. We find that for a z-score of -1, the cumulative probability is 0.1587, and for a z-score of 1, the cumulative probability is 0.8413.

Finally, we subtract the lower probability from the higher probability to find the probability of the heights between 169 and 181 cm:
Probability = 0.8413 - 0.1587 = 0.6826

To find the number of people expected to fall within this range, we multiply the probability by the total number of people:
Number of people = Probability * Total number of people = 0.6826 * 40 = 27.304 ≈ 27

Therefore, we would expect approximately 27 people to be between 169 and 181 cm tall.

(b) To find the number of people expected to be taller than 170 cm, we need to calculate the probability of a height being greater than 170 cm.

First, we calculate the z-score for a height of 170 cm:
z = (170 - mean) / standard deviation = (170 - 175) / 6 = -0.8333

Next, we find the cumulative probability for a z-score of -0.8333 from the standard normal distribution table. The cumulative probability is 0.2033.

Finally, we subtract this probability from 1 to find the probability of being taller than 170 cm:
Probability = 1 - 0.2033 = 0.7967

To find the number of people expected to be taller than 170 cm, we multiply the probability by the total number of people:
Number of people = Probability * Total number of people = 0.7967 * 40 = 31.868 ≈ 32

Therefore, we would expect approximately 32 people to be taller than 170 cm.