The height , in meters, of 500 people are normally distributed with standard deviation 0.080.Given that the heights of 129 of these people are greater than the mean height, but less than 1.806m,estimate the mean height.
To estimate the mean height of the 500 people, we can use the information provided: 129 individuals have heights greater than the mean but less than 1.806m, and we know that the standard deviation is 0.080.
First, let's calculate the z-score for the given height of 1.806m using the formula:
z = (x - μ) / σ
where:
- z is the z-score
- x is the given height of 1.806m
- μ is the mean height (which we want to estimate)
- σ is the standard deviation of 0.080
Substituting the given values:
z = (1.806 - μ) / 0.080
Next, we determine the area to the left of this z-score using a standard normal distribution table. The area represents the percentage of values below this z-score.
From the table, we find that the area to the left of the z-score corresponding to 1.806 is approximately 0.9641.
Since we are given that 129 individuals fall within this range, the proportion of the total population can be calculated:
proportion = 129 / 500 = 0.258
Now, we need to find the z-score that corresponds to this proportion using the same standard normal distribution table. The closest value we find is 0.6042.
We can now set up an equation using the z-scores for the given height and the calculated proportion:
0.9641 - 0.6042 = (1.806 - μ) / 0.080
Simplifying the equation gives:
0.3599 = (1.806 - μ) / 0.080
To solve for μ, we multiply both sides of the equation by 0.080:
0.080 * 0.3599 = 1.806 - μ
0.0288 = 1.806 - μ
Rearranging the equation to isolate μ, we have:
μ = 1.806 - 0.0288
μ ≈ 1.7772
Therefore, the estimated mean height of the 500 people is approximately 1.7772 meters.
If you sketch out your normal curve you are looking at the shaded section above the mean but below 1.806
And remember your z-score table reads less than... so you will need to take 1 - your section number : )