Question

The brain weights of a certain population of adult Swedish males follow approximately a normal distribution with mean \(1,400 \mathrm{gm}\) and standard deviation \(100 \mathrm{gm} .\) What percentage of the brain weights are (a) \(1,500 \mathrm{gm}\) or less? (b) between 1,325 and \(1,500 \mathrm{gm} ?\) (c) 1,325 gm or more? (d) 1,475 gm or more? (e) between 1,475 and \(1,600 \mathrm{gm} ?\) (f) between 1,200 and \(1,325 \mathrm{gm} ?\)

Short answer

(a) 84.13%, (b) 68.26%, (c) 95.45%, (d) 31.73%, (e) 13.59%, (f) 4.55%.

Step-by-step solution

Standardize the values

Convert the brain weights to z-scores using the formula: \( z = \frac{X - \mu}{\sigma} \), where \(X\) is the value, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.

Calculate z-scores for each value

Calculate the z-scores for the values given in the problems: \( z_{1500} = \frac{1500 - 1400}{100} \), \( z_{1325} = \frac{1325-1400}{100} \), \( z_{1475} = \frac{1475-1400}{100} \), \( z_{1600} = \frac{1600-1400}{100} \), \( z_{1200} = \frac{1200-1400}{100} \).

Use z-table to find corresponding probabilities

Use the standard normal distribution table to find the probabilities that correspond to the calculated z-scores.

Calculate the required percentages

For (a), look up the probability for \( z_{1500} \). For (b), find the probabilities for both \( z_{1325} \) and \( z_{1500} \) and subtract. For (c), calculate it as 1 - probability of \( z_{1325} \). For (d), it's 1 - probability of \( z_{1475} \). For (e), find the probability difference between \( z_{1475} \) and \( z_{1600} \). For (f), calculate the difference between the probabilities for \( z_{1200} \) and \( z_{1325} \).

Key concepts

Z-Score Calculation

In statistics, the z-score is an incredibly useful tool, helping us understand how a specific data point compares to the rest of the data set. A z-score reveals how many standard deviations a value is from the mean of the data set.

To calculate the z-score, you use the formula:
\( z = \frac{X - \mu}{\sigma} \), where \( X \) represents the value in question, \( \mu \) is the mean, and \( \sigma \) is the standard deviation of the data set. If the z-score is 0, it implies that the data point is identical to the mean. Positive z-scores indicate values above the mean, while negative z-scores suggest values below the mean.

For instance, if you wish to calculate the z-score for a brain weight of 1500 gm in our exercise, you would subtract the mean (1400 gm) from 1500 gm and divide by the standard deviation (100 gm), giving you a z-score of 1. This means that 1500 gm is one standard deviation above the mean brain weight.

Standard Deviation

The concept of standard deviation is pivotal in the realm of statistics. It's the measure of the amount of variation or dispersion present in a set of values. A lower standard deviation means that the values are close to the mean, while a higher standard deviation indicates that the values are more spread out.

Standard deviation is represented by \( \sigma \) and calculates the average distance from the mean for all the data points in the set. You can think of it as an 'average' deviation from the mean. Mathematically, it involves squaring the difference between each data point and the mean, averaging those squares, and finally taking the square root. It is an essential measure because it gives context to the mean, providing insight into the variability of the data set.

Probability Calculation

Probability calculation in a normal distribution allows us to determine the likelihood of a particular value occurring within a dataset. To find this probability, after calculating the z-score, we look it up in the standard normal distribution table, often referred to as a z-table.

The z-table shows us the probability of a z-score being less than or equal to a certain value. To express the probability of a z-score being greater than a certain point, we simply subtract the value from 1 since the total area under the normal distribution curve is 1, or 100%.

For example, if you're looking to calculate the probability of a brain weight of 1500 gm or less, you find the z-score as in the previous step, then use the z-table to find the probability that corresponds to this z-score. This reflects the percentage of brain weights falling at or below 1500 gm.

Normal Distribution Table

The normal distribution table, also known as the z-table, is an integral tool for working with data that is normally distributed. It allows us to translate z-scores into probabilities. Because the normal distribution is symmetrical, the table reflects the probability that a z-score falls within a certain leftward area under the curve.

Typically, the table provides the probability that a z-score is less than a given value. The table is split in the middle with positive z-scores on one side and negative z-scores on the other. To use it, you first calculate your z-score and then find the corresponding probability in the table.

For rarer events, which have z-scores not listed in the z-table, statistical software or more extensive tables can be used. Understanding and interpreting this table is a critical skill for any statistics student, as it is extensively used in fields ranging from psychology to finance.