Problem 39 - Basic Statistics for Business and Economics with Student

Question

The mean SAT score for Division I student-athletes is 947 with a standard deviation of \(205 .\) If you select a random sample of 60 of these students, what is the probability the mean is below \(900 ?\)

Step-by-Step Solution

Answer

The probability is approximately 3.84%.

Step 1: Identify the Given Values

We have the mean SAT score for Division I student-athletes, \( \mu = 947 \), and the standard deviation, \( \sigma = 205 \). The sample size \( n = 60 \). We need to find the probability that the sample mean, \( \bar{x} \), is below 900.

Step 2: Calculate the Standard Error of the Mean

The standard error of the mean (SEM) is calculated using the formula: \( \text{SEM} = \frac{\sigma}{\sqrt{n}} \). Substituting the given values, \( \text{SEM} = \frac{205}{\sqrt{60}} \approx 26.47 \).

Step 3: Compute the Z-score

We find the Z-score for the sample mean \( \bar{x} = 900 \) using the formula: \( Z = \frac{\bar{x} - \mu}{\text{SEM}} \). Substituting the values, \( Z = \frac{900 - 947}{26.47} \approx -1.77 \).

Step 4: Find the Probability using the Z-table

Using the Z-table, we look up the probability corresponding to \( Z = -1.77 \). This gives us the probability \( P(Z < -1.77) \approx 0.0384 \).

Step 5: Conclude the Result

The probability that the sample mean SAT score of these student-athletes is below 900 is approximately 0.0384, or 3.84%.

Key Concepts

Standard Deviation

To fully understand the concept of the standard deviation, let's think about variability within data. It’s all about measuring how spread out the numbers are from the mean. In our example, the mean SAT score is 947. But not everyone scores exactly 947, right? Some do better, and some do worse. The standard deviation, which in this case is 205, gives us an idea of how much scores deviate from this average on either side.

• A low standard deviation indicates that the values are clustered close to the mean.
• A high standard deviation indicates a wide spread of values.

This concept is crucial when dealing with probability and inferential statistics because it helps us determine how typical or atypical a score is. In our example, knowing the SAT score standard deviation tells us how much variation to expect from one student-athlete compared to another.

Standard Error of the Mean

The standard error of the mean (often abbreviated as SEM) is crucial for understanding how accurately a sample mean estimates the true population mean. It's vital when you have a sample, like our group of 60 student-athletes, rather than the entire population. The formula to calculate SEM is:\[ SEM = \frac{\sigma}{\sqrt{n}} \]where:

• \( \sigma \) is the standard deviation of the population,
• \( n \) is the sample size.

For our exercise, the calculation yielded around 26.47. This means our sample mean’s accuracy in estimating the true mean of all student-athletes is roughly within 26.47 points. A smaller SEM implies a more accurate sample mean, which is why larger samples give us more reliable estimates.

Z-score Calculation

When you hear about a Z-score, think of it as a way to figure out how far away a particular value is from the mean in terms of standard deviations. This is particularly useful when comparing different data points or for assessing probabilities. The formula to determine a Z-score is:\[ Z = \frac{\bar{x} - \mu}{SEM} \]where:

• \( \bar{x} \) is the sample mean,
• \( \mu \) is the population mean,
• SEM is the standard error of the mean.

For the SAT problem, we computed a Z-score of approximately -1.77. This Z-score suggests that the sample mean of 900 is 1.77 standard deviations below the population mean. By referring to Z-tables, this helps us discover probabilities, providing insight into how likely it is to obtain such a sample mean. In this scenario, it tells us there’s about a 3.84% chance the student sample mean score will fall below 900.