Assume that women’s heights are normally distributed with a mean of 63.6inches and a standard deviation of 2.5inches. If 90 women are randomly selected, find the probability that they have a mean height between 62.9inches and 64.0inches.
A nice online Z-table gui can be found at
http://davidmlane.com/hyperstat/z_table.html
To find the probability that the mean height of 90 women is between 62.9 inches and 64.0 inches, we need to use the Central Limit Theorem, which states that the distribution of sample means will be approximately normally distributed regardless of the shape of the original population, as long as the sample size is large enough.
In this case, we are given that the population mean µ = 63.6 inches and the population standard deviation σ = 2.5 inches.
First, we need to calculate the standard error of the mean (SE), which is given by the formula:
SE = σ / √(n),
where σ is the population standard deviation and n is the sample size.
In this case, the sample size is n = 90, so the standard error of the mean is:
SE = 2.5 / √(90) ≈ 0.2632 inches.
Next, we need to standardize the values of 62.9 inches and 64.0 inches using the formula:
Z = (X - µ) / SE,
where X is the given value, µ is the population mean, and SE is the standard error of the mean.
For 62.9 inches:
Z1 = (62.9 - 63.6) / 0.2632 ≈ -2.6598.
For 64.0 inches:
Z2 = (64.0 - 63.6) / 0.2632 ≈ 1.522 .
Now, we can use the Z-table or calculator to find the probability that the mean height is between -2.6598 and 1.522. We need to find the area under the standard normal curve between these two Z-values.
Using the Z-table, we can find that the area to the left of -2.6598 is approximately 0.0039, and the area to the left of 1.522 is approximately 0.9357.
Therefore, the probability that the mean height of 90 women is between 62.9 inches and 64.0 inches is:
P(-2.6598 < Z < 1.522) ≈ 0.9357 - 0.0039 ≈ 0.9318, or approximately 93.18%.
To find the probability that the mean height of the 90 randomly selected women falls between 62.9 inches and 64.0 inches, we can use a standard normal distribution.
Step 1: Convert the individual height of 62.9 inches and 64.0 inches to z-scores.
The z-score is calculated using the formula:
z = (x - μ) / σ
For 62.9 inches:
z1 = (62.9 - 63.6) / 2.5
z1 = -0.28
For 64.0 inches:
z2 = (64.0 - 63.6) / 2.5
z2 = 0.16
Step 2: Look up the cumulative probability associated with the z-scores.
We can use the Z-table or a statistical calculator to find the cumulative probability for each z-score.
For z1 = -0.28, the cumulative probability is approximately 0.3907.
For z2 = 0.16, the cumulative probability is approximately 0.5636.
Step 3: Calculate the probability between the two z-scores.
To find the probability that the mean height falls between 62.9 inches and 64.0 inches, we subtract the cumulative probability of the lower z-score from the cumulative probability of the higher z-score.
P(62.9 ≤ X ≤ 64.0) = P(z1 ≤ Z ≤ z2) = P(Z ≤ z2) - P(Z ≤ z1)
P(62.9 ≤ X ≤ 64.0) = 0.5636 - 0.3907
P(62.9 ≤ X ≤ 64.0) ≈ 0.1729
Therefore, the probability that the mean height of the 90 randomly selected women falls between 62.9 inches and 64.0 inches is approximately 0.1729 or 17.29%.
You'd need your graphing calculator for this
Go to "distr" and click "normalcdf("
Type this in ⤵︎
lower: 62.9
upper: 64
µ: 63.6
σ: 2.5
Click "Paste" and you'll get your answer