Assume that the heights of American women are approximately normally distributed with a mean of 66.5 in. and a standard deviation of 2.5 in. Within what range are the heights of 95% of American women? Please help immediately!!! :(
Z = ±1.96 for 95%
Z = (score-mean)/SD
Insert values and calculate.
To find the range within which 95% of American women's heights fall, we need to use the concept of Z-scores and the standard normal distribution.
Here's how you can approach this problem step-by-step:
Step 1: Identify the Z-score for the desired percentage of data.
Given that we want to find the range that contains 95% of the data, we need to find the corresponding Z-score, which can be obtained from the standard normal distribution table or by using a statistical calculator.
The Z-score associated with a 95% probability is approximately equal to 1.96.
Step 2: Use the Z-score formula to find the range.
The Z-score formula is:
Z = (X - μ) / σ
where:
- Z is the Z-score
- X is the raw score (height in this case)
- μ is the mean of the distribution
- σ is the standard deviation of the distribution
Since we want to calculate the range within which 95% of American women's heights fall, we need to find the highest and lowest values that correspond to the Z-scores associated with 1.96.
Step 3: Calculate the highest and lowest values.
For the highest value, we can set up the equation as follows:
1.96 = (X - 66.5) / 2.5
Solving for X:
1.96 * 2.5 = X - 66.5
4.9 + 66.5 = X
X = 71.4
For the lowest value, we can set up the equation as follows:
-1.96 = (X - 66.5) / 2.5
Solving for X:
-1.96 * 2.5 = X - 66.5
-4.9 + 66.5 = X
X = 61.6
Step 4: Determine the range.
Based on our calculations, 95% of American women's heights fall within the range of 61.6 inches to 71.4 inches.
So, to answer your question, the range within which 95% of American women's heights are expected to be found is between 61.6 inches and 71.4 inches.