Question

Companies that design furniture for elementary school classrooms produce a variety of sizes for kids of different ages. Suppose the heights of kindergarten children can be described by a Normal model with a mean of \(38.2\) inches and standard deviation of \(1.8\) inches. a) What fraction of kindergarten kids should the company expect to be less than 3 feet tall? b) In what height interval should the company expect to find the middle \(80 \%\) of kindergarteners? c) At least how tall are the biggest \(10 \%\) of kindergarteners?

Short answer

a) 11.12% b) 35.9 to 40.5 inches c) At least 40.5 inches.

Step-by-step solution

Convert Feet to Inches for Part (a)

We need to find out how tall 3 feet is in inches. Since 1 foot equals 12 inches, 3 feet equals:\[ 3 \text{ feet} \times 12 \text{ inches/foot} = 36 \text{ inches} \]

Standardize the Height for Part (a)

To find the probability of a child being less than 36 inches tall, we standardize using the Z-score formula:\[ Z = \frac{X - \mu}{\sigma} \]Where \(X = 36\), \(\mu = 38.2\), and \(\sigma = 1.8\):\[ Z = \frac{36 - 38.2}{1.8} \approx -1.222 \]

Find Probability for Part (a)

Using the standard normal distribution table, find the probability that a child's height is less than 36 inches (\(Z < -1.222\)). This corresponds to approximately 0.1112, or 11.12% of kindergarten kids.

Find Z-Scores for 80% Interval in Part (b)

For the middle 80% of kindergarteners, we need to find the 10th and 90th percentiles of the standard normal distribution. These values correspond approximately to \(Z = -1.28\) and \(Z = 1.28\).

Convert Z-Scores to Heights for Part (b)

Convert the Z-scores back to height values using the formula:\[ X = Z \cdot \sigma + \mu \]For \(Z = -1.28\):\[ X = -1.28 \cdot 1.8 + 38.2 \approx 35.9 \text{ inches} \]For \(Z = 1.28\):\[ X = 1.28 \cdot 1.8 + 38.2 \approx 40.5 \text{ inches} \]So, the middle 80% interval is from 35.9 to 40.5 inches.

Find Z-Score for Top 10% in Part (c)

To find how tall the biggest 10% of kindergarteners are, find the Z-score that corresponds to the 90th percentile.This Z-score is approximately \(Z = 1.28\).

Convert Z-Score to Height for Part (c)

Convert the Z-score to a height using the formula:\[ X = Z \cdot \sigma + \mu \]For \(Z = 1.28\):\[ X = 1.28 \cdot 1.8 + 38.2 \approx 40.5 \text{ inches} \]Therefore, the tallest 10% of kindergarteners are at least 40.5 inches tall.

Key concepts

Standard Deviation

The standard deviation is a measure that tells us how spread out the data is in a normal distribution. In simpler terms, it helps us understand how much the values in a data set deviate from the average (mean) value. For example, in the context of our exercise, the average height of kindergarten children is 38.2 inches, and the standard deviation is 1.8 inches. This means:
- Most kindergarten children have a height around 38.2 inches.
- The heights usually differ from this average by about 1.8 inches either more or less.
When standard deviation is small, data points are clustered close around the average. If it's large, the data points are more spread out. Understanding the standard deviation gives us a clearer picture of the variability in students' heights, which can be practical for designing furniture that accommodates most children comfortably.

Z-score

The Z-score is a valuable tool in statistics for identifying how far away a particular data point is from the mean, expressed in terms of standard deviations. It’s calculated using the formula:\[ Z = \frac{X - \mu}{\sigma} \]Where:

• \(X\) is the value in question.
• \(\mu\) is the mean of the data set.
• \(\sigma\) is the standard deviation.

For example, to find how many kindergarten kids are shorter than 3 feet (36 inches), we calculate the Z-score for 36 inches. If the Z-score is negative, the height is below the average. By using the standard normal distribution table, we can look up the corresponding probability for a Z-score, helping us understand how common or rare a particular data point is. In this problem, a Z-score of around -1.222 indicates that 36 inches is below average, and only about 11.12% of kids would be shorter than this height.

Percentiles

Percentiles help us understand the relative standing of a measurement within a data set. They indicate the position of a value in a distribution relative to all other values. For example, saying that a child's height is in the 10th percentile means that they are taller than 10% of their peers and shorter than 90% of them. In this exercise, to find the height range where the middle 80% of kindergarten children fall, we identify the 10th and 90th percentiles. These percentiles are directly related to Z-scores, with about -1.28 for the 10th percentile and 1.28 for the 90th percentile.

• The middle 80% of heights fall between these Z-scores when converted back to the original data scale, which in this case translates to heights between 35.9 inches and 40.5 inches.
• The tallest 10% of children are those whose heights exceed the 90th percentile, highlighting that tall furniture might be necessary for these kids, who would typically be at least 40.5 inches tall.

Understanding percentiles can guide us in making better decisions, like designing furniture that fits most children well.