18. If the mean test score of a distribution of exam scores is 78.75 with a standard deviation of 4.95, then determine the test score (to 2 decimal places) of a student that had a z score of -1.92? a) 88.25 b) 69.25 c) 76.83 d) 73.80 cannot be estimated
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
![**Question 18:** If the mean test score of a distribution of exam scores is 78.75 with a standard deviation of 4.95, then determine the test score (to 2 decimal places) of a student that had a z-score of -1.92?
**Options:**
a) 88.25
b) 69.25
c) 76.83
d) 73.80
e) cannot be estimated
**Solution:**
To determine the test score corresponding to a z-score of -1.92, we use the formula for z-score:
\[ z = \frac{(X - \mu)}{\sigma} \]
where:
- \( z \) is the z-score
- \( X \) is the test score
- \( \mu \) is the mean of the test scores
- \( \sigma \) is the standard deviation
Given:
- \( z = -1.92 \)
- \( \mu = 78.75 \)
- \( \sigma = 4.95 \)
We rearrange the formula to solve for \( X \):
\[ X = \mu + z \cdot \sigma \]
Substitute the given values:
\[ X = 78.75 + (-1.92) \cdot 4.95 \]
\[ X = 78.75 - 9.504 \]
\[ X = 69.246 \]
Rounded to two decimal places, the test score is 69.25.
Therefore, the correct answer is:
**b) 69.25**](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8c4811d0-c10d-4cc4-a20d-9fe861f3e2a2%2F9f852926-149a-416a-8b97-405d3f962619%2F1atj8co.png&w=3840&q=75)
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