Suppose that people's heights (in centimeters) are normally distributed, with a mean of 170 and a standard deviation of 5. We find the heights of 50 people. (You may need to use the standard normal distribution table. Round your answers to the nearest whole number.)
(a) How many would you expect to be between 170 and 180 cm tall? =29
b How many would be taller than 176 cm=7
To answer these questions, we need to use the properties of the normal distribution.
(a) To find the number of people expected to be between 170 and 180 cm tall, we first need to calculate the z-scores for these heights. The z-score formula is:
z = (x - μ) / σ
where x is the value we want to find the z-score for, μ is the mean of the distribution, and σ is the standard deviation.
For the lower bound of 170 cm:
z_lower = (170 - 170) / 5 = 0
For the upper bound of 180 cm:
z_upper = (180 - 170) / 5 = 2
Next, we need to consult a standard normal distribution table or use a calculator to find the area under the standard normal curve between these two z-scores. This area represents the proportion of the population with heights between 170 and 180 cm.
From the standard normal distribution table, we find that the area between z = 0 and z = 2 is approximately 0.4772.
Finally, we multiply the proportion by the total number of people to find the expected number between 170 and 180 cm:
Expected number = 0.4772 * 50 = 23.86
Since we are rounding to the nearest whole number, the expected number of people between 170 and 180 cm tall is 24.
(b) To find the number of people taller than 176 cm, we need to calculate the z-score for 176 cm using the same formula:
z = (x - μ) / σ
z = (176 - 170) / 5
z = 6 / 5
z = 1.2
Next, we need to find the area under the standard normal curve to the right of z = 1.2. From the standard normal distribution table, we find that the area to the right of z = 1.2 is approximately 0.1151.
Finally, we multiply the proportion by the total number of people to find the expected number taller than 176 cm:
Expected number = 0.1151 * 50 = 5.76
Since we are rounding to the nearest whole number, the expected number of people taller than 176 cm is 6.
To find the number of people expected to be between 170 and 180 cm tall, we need to calculate the area under the normal distribution curve between these two values.
(a)
1. Convert the given values to z-scores using the formula: z = (x - μ) / σ, where x is the given value, μ is the mean, and σ is the standard deviation.
For 170 cm:
z = (170 - 170) / 5 = 0
For 180 cm:
z = (180 - 170) / 5 = 2
2. Use the standard normal distribution table to find the area between 0 and 2 under the curve.
From the table, the area corresponding to a z-score of 0 is 0.5000, and the area corresponding to a z-score of 2 is 0.9772.
3. Subtract the area corresponding to a z-score of 0 from the area corresponding to a z-score of 2 to find the area between 0 and 2.
0.9772 - 0.5000 = 0.4772
4. Multiply the obtained area by the total number of people (50) to find the expected number of people between 170 and 180 cm tall.
0.4772 * 50 = 23.86 (rounded to the nearest whole number) = 24
Therefore, we would expect approximately 24 people to be between 170 and 180 cm tall.
(b)
1. Convert the given value of 176 cm to a z-score.
For 176 cm:
z = (176 - 170) / 5 = 1.2
2. Use the standard normal distribution table to find the area to the right of a z-score of 1.2 (which represents being taller than 176 cm).
From the table, the area corresponding to a z-score of 1.2 is 0.3849.
3. Multiply the obtained area by the total number of people (50) to find the expected number of people taller than 176 cm.
0.3849 * 50 = 19.245 (rounded to the nearest whole number) = 19
Therefore, we would expect approximately 19 people to be taller than 176 cm.
Z = (score-mean)/SD = (180-170)/5 = 2
a. Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability between score and mean for Z score. Multiply by 50 ≠ 29
b. Z = (176-170)/5 = 6/5 = 1.2
Smaller portion multiplied by 50 ≠ 7