The distribution of students’ heights in a class of 100 students is normal, with a mean

height of 66 inches and a standard deviation of three. With these parameters, answer the associated question(s).
The shortest 10% of the class is equal to or shorter than __________ inches. Round to the nearest tenths place if a fraction.

Z = (score-mean)/SD

Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability (.10) in the smaller area and find the Z score. Insert in equation above to find the score.

To find the height that corresponds to the shortest 10% of the class, we need to calculate the z-score associated with this percent. The z-score represents the number of standard deviations away a particular value is from the mean.

To find the z-score, we can use the z-score formula:
z = (x - μ) / σ

Where:
x = the value we want to find the z-score for (in this case, the lowest 10%)
μ = the mean height (66 inches)
σ = the standard deviation (3 inches)

To find the z-score for the lowest 10% of the class, we need to find the z-score that corresponds to a cumulative area of 0.10, or 10% (because the normal distribution is symmetric). We can look up this value in a z-table or use a calculator.

Using a z-table or calculator, the z-score corresponding to 10% is approximately -1.28.

Now, we can solve for the value of x (the shortest 10% of the class):
-1.28 = (x - 66) / 3

Rearrange the equation to solve for x:
x - 66 = -1.28 * 3
x - 66 = -3.84
x = -3.84 + 66
x ≈ 62.16

Therefore, the shortest 10% of the class is equal to or shorter than approximately 62.2 inches (rounded to the nearest tenth).

To find the height that corresponds to the shortest 10% of the class, we need to use the concept of z-scores in a normal distribution.

A z-score measures the number of standard deviations a particular data point is from the mean. It can be calculated using the formula:

z = (x - μ) / σ

Where:
- x is the data point
- μ is the mean of the distribution
- σ is the standard deviation of the distribution

In this case, we want to find the height that corresponds to the shortest 10%. To do this, we need to find the z-score associated with the 10th percentile.

Since the distribution is normal, we know that the area under the curve to the left of the z-score is equal to the percentile we're interested in (10%). We can use a standard normal distribution table or a statistical calculator to find the z-score that corresponds to the 10th percentile.

Using the table or calculator, we find that the z-score corresponding to the 10th percentile is approximately -1.28.

Now we can use the z-score formula to find the corresponding height:

-1.28 = (x - 66) / 3

Rearranging the formula, we get:

x = (-1.28 * 3) + 66

Calculating this expression, we find:

x ≈ 62.16 inches

So, the shortest 10% of the class is equal to or shorter than approximately 62.16 inches.

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