a sample of men's heights was taken. The average height was 68.5 inches, the SD was 1.95 inches. Use the normal curve to estimate the percentage of men with heights between 69 inches and 70 inches.
I use this fantastic webpage for these kind of questions
http://davidmlane.com/hyperstat/z_table.html
enter the mean and SD, click on "between" and enter 69 and 70 to get
.1779
To estimate the percentage of men with heights between 69 inches and 70 inches using the normal curve, we can use the concept of Z-scores and the standard normal distribution.
First, let's calculate the Z-scores for the given data.
Z-score formula: Z = (X - μ) / σ
Where:
- X is the value we want to calculate the Z-score for (69 and 70 inches in this case).
- μ is the mean (average) height (68.5 inches in this case).
- σ is the standard deviation (SD) (1.95 inches in this case).
For the height of 69 inches:
Z1 = (69 - 68.5) / 1.95 = 0.2564
For the height of 70 inches:
Z2 = (70 - 68.5) / 1.95 = 0.7692
Next, we can use a Z-table or a calculator to find the probabilities associated with these Z-scores.
Using a Z-table, we can find the corresponding probabilities.
P(Z < 0.2564) = 0.5984 (rounded to four decimal places)
P(Z < 0.7692) = 0.7772 (rounded to four decimal places)
Now, we can calculate the percentage of men with heights between 69 inches and 70 inches by subtracting the lower probability from the higher probability.
P(69 < X < 70) = P(Z < 0.7692) - P(Z < 0.2564)
P(69 < X < 70) = 0.7772 - 0.5984
P(69 < X < 70) ≈ 0.1788
Therefore, the estimated percentage of men with heights between 69 inches and 70 inches is approximately 17.88%.