A set of final examination grades in an introductory statistics course is normally​ distributed, with a mean of 72 and a standard deviation of 9. What is the probability that a student scored below 88 on this​ exam?

To find the probability that a student scored below 88 on the exam, we need to compute the cumulative probability for a normally distributed variable. 

Given:
- Mean (\(\mu\)) = 72
- Standard deviation (\(\sigma\)) = 9
- We need the probability that a student scored below 88, i.e., \(P(X < 88)\).

To solve this, we use the Z-score formula:

\[
Z = \frac{X - \mu}{\sigma}
\]

where \(X = 88\), \(\mu = 72\), and \(\sigma = 9\).

Let's calculate the Z-score first:

\[
Z = \frac{88 - 72}{9} = \frac{16}{9} \approx 1.78
\]

Next, we find the cumulative probability corresponding to this Z-score using standard normal distribution tables or a calculator.

The cumulative probability for a Z-score of 1.78 is approximately 0.9625.

Thus, the probability that a student scored below 88 is approximately **96.25%**.

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Would you like further details or clarifications?

Here are some related questions you might find helpful:

1. What is the probability of a student scoring above 88?
2. How do you find the probability of a student scoring between 70 and 88?
3. How do you calculate the probability for any other score, say 60, using Z-scores?
4. What is the Z-score for a student scoring exactly 72?
5. How do you interpret Z-scores and cumulative probabilities in normal distributions?

**Tip:** The Z-score represents how many standard deviations a value is from the mean. A positive Z-score indicates a value above the mean, while a negative Z-score indicates a value below the mean.

A set of final examination grades in an introductory statistics course is normally distributed, with a mean of 72 and a standard deviation of 9. What is the probability that a student scored below 88 on this exam?

This problem explores how to calculate the probability of a student scoring below 88 in an exam, assuming a normal distribution with a mean of 72 and a standard deviation of 9. Learn how to apply the Z-score formula and use standard normal distribution tables to find probabilities in introductory statistics.

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