The heights of 200 kindergarten students are normally distributed with a mean of 40 and a standard deviation of 1.8 inches. Approximately how many students have a height between 37.3 inches and 44.5 inches?
To find the approximate number of students with a height between 37.3 inches and 44.5 inches, we need to calculate the z-scores for both values and then find the area between these z-scores under the normal distribution curve.
The z-score formula is given by:
z = (x - mean) / standard deviation
For the lower value of 37.3 inches:
z1 = (37.3 - 40) / 1.8
For the upper value of 44.5 inches:
z2 = (44.5 - 40) / 1.8
Calculating the z-scores:
z1 ≈ -1.472
z2 ≈ 2.472
Using a standard normal distribution table or a statistical software, we can find the area between these z-scores.
Looking up the z-scores in the standard normal distribution table, we can find the following probabilities:
P(z < -1.472) ≈ 0.0708 (Using a z-table or a calculator)
P(z < 2.472) ≈ 0.9924
To find the area between these two probabilities:
P(-1.472 < z < 2.472) = P(z < 2.472) - P(z < -1.472)
≈ 0.9924 - 0.0708
≈ 0.9216
This means that approximately 92.16% of the students have a height between 37.3 inches and 44.5 inches.
To find the approximate number of students within this range, we can multiply the percentage by the total number of students:
Approximate number of students = 0.9216 * 200
≈ 184.32
Therefore, approximately 184 students have a height between 37.3 inches and 44.5 inches.
To find the approximate number of students with a height between 37.3 inches and 44.5 inches, we can use the concept of standard deviation and the properties of the normal distribution.
1. Start by calculating the z-scores for the lower and upper limits of the height range. The z-score measures the number of standard deviations an observation is from the mean.
For the lower limit (37.3 inches):
z = (37.3 - 40) / 1.8
For the upper limit (44.5 inches):
z = (44.5 - 40) / 1.8
2. Use a standard normal distribution table (also known as a z-table) or a statistical software to find the area under the standard normal curve between these two z-scores.
Alternatively, you can use a calculator or an online tool that provides the cumulative probability of a normal distribution given the z-scores.
3. Take the difference between these two areas to find the approximate proportion of students with a height between 37.3 inches and 44.5 inches.
4. Finally, multiply this proportion by the total number of students (200) to estimate the number of students within the given height range.
It is important to note that the "approximately" in the question implies that we are using an approximation because the normal distribution is continuous and the actual number of students falling within the range will be a whole-number count.
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 between the two Z scores. Multiply that by 200.