Question

Mathematics achievement test scores for 400 students had a mean and a variance equal to 600 and \(4,900,\) respectively. If the distribution of test scores was mound-shaped, approximately how many scores would fall in the interval 530 to 670 ? Approximately how many scores would fall in the interval 460 to \(740 ?\)

Short answer

Answer: Approximately 273 test scores would fall in the interval 530 to 670, and 382 scores would fall in the interval 460 to 740.

Step-by-step solution

Calculate the standard deviation

To find the standard deviation, we will use the formula \(\sigma = \sqrt{variance}.\) We know that the variance of this data set is \(4,900.\) So, the standard deviation is: \(\sigma = \sqrt{4900} \approx 70.\)

Calculate z-scores

Now, we need to find the z-scores for the interval bounds. The formula for z-scores is: \(z = \frac{x - \mu}{\sigma},\) where \(x\) is the data value, \(\mu\) is the mean, and \(\sigma\) is the standard deviation. Since the mean is 600 and the standard deviation is 70, we have: 1. Interval 530 to 670: \(z_{530} = \frac{530 - 600}{70} \approx -1,\) \(z_{670} = \frac{670 - 600}{70} \approx 1.\) 2. Interval 460 to 740: \(z_{460} = \frac{460 - 600}{70} \approx -2,\) \(z_{740} = \frac{740 - 600}{70} \approx 2.\)

Find the proportion of scores in each interval

Next, we need to find the proportion of scores that fall in each interval using the z-scores. We will use the standard normal distribution table for this purpose. 1. Interval 530 to 670: Proportion = \(P(-1 < z < 1) \approx 0.6827.\) 2. Interval 460 to 740: Proportion = \(P(-2 < z < 2) \approx 0.9545.\)

Find the number of scores in each interval

Finally, we need to multiply the proportion of scores for each interval by the total number of students (400) to find how many scores fall in each interval: 1. Interval 530 to 670: Number of scores = \(0.6827 * 400 \approx 273.\) 2. Interval 460 to 740: Number of scores = \(0.9545 * 400 \approx 382.\) So, approximately 273 scores would fall in the interval 530 to 670, and 382 scores would fall in the interval 460 to 740.

Key concepts

Standard Deviation Calculation

Standard deviation is a crucial statistic that measures the amount of variation or dispersion in a set of values. It indicates how spread out the numbers are in a data set, and higher standard deviation means more spread.

To calculate standard deviation, you first need to compute the variance, which is the average of the squared differences from the mean. Then, the standard deviation is simply the square root of the variance. In mathematical terms, if the variance is denoted by \(\sigma^2\), the standard deviation \(\sigma\) is calculated as \(\sigma = \sqrt{\sigma^2}\).

Understanding standard deviation is critical because it's used in calculating z-scores, which compare individual data points to the mean. In the aforementioned exercise, the standard deviation is found to be approximately 70, derived from the given variance of 4900. This calculation is foundational for interpreting how far specific values lie from the mean and for further statistical analysis.

Normal Distribution

The normal distribution, also known as the Gaussian distribution, is a symmetric, bell-shaped distribution that depicts how the values of a variable are distributed. It is defined by two parameters: the mean \(\mu\), which is the peak of the curve and represents the average of the distribution; and the standard deviation \(\sigma\), which determines the width of the curve.

A key property of the normal distribution is that it’s completely described by the mean and standard deviation. Approximately 68% of data within one standard deviation, 95% within two, and 99.7% within three standard deviations from the mean. This proportion remains consistent regardless of the mean or standard deviation of the distribution, which is known as the Empirical Rule.

In the example problem, we are told the distribution is ‘mound-shaped’, which often implies a normal distribution, allowing us to use the characteristics of the normal curve to estimate the number of students' scores that fall within certain intervals.

Probability and Statistics

Probability and statistics are interconnected fields that deal with uncertainties and the quantification of data. Probability is the measure of the likelihood that an event will occur, while statistics is the science concerned with the collection, analysis, interpretation, and presentation of data.

When it comes to applying probability to statistics, one common method is to use a z-score table, also known as a standard normal distribution table. This table provides the probability that a statistic is observed below, above, or between certain standard deviations from the mean in a normal distribution.

In the context of the textbook problem, after calculating z-scores for specific intervals, these values are looked up in a z-score table to determine the proportion of the total area under the curve that lies within these intervals. Multiplying these proportions by the total number of students gives us the estimations needed: approximately 273 students' scores fall between 530 and 670, and roughly 382 students' scores lie between 460 and 740.