Lisa has an average bowling score of 190 in her winter league where the average for all bowlers is 175 with a standard deviation is 9. Barb has an average bowling score of 175 in the spring league where the average for all bowlers is 155 with a standard deviation of 10. (a) What is Lisa's z score? Round your answer to two decimal places as necessary. (b) What is Barb's z score? Round your answer to two decimal places as necessary. (c) Who scores higher in their own league if both league scores are normally distributed?
Inverse Normal Distribution
The method used for finding the corresponding z-critical value in a normal distribution using the known probability is said to be an inverse normal distribution. The inverse normal distribution is a continuous probability distribution with a family of two parameters.
Mean, Median, Mode
It is a descriptive summary of a data set. It can be defined by using some of the measures. The central tendencies do not provide information regarding individual data from the dataset. However, they give a summary of the data set. The central tendency or measure of central tendency is a central or typical value for a probability distribution.
Z-Scores
A z-score is a unit of measurement used in statistics to describe the position of a raw score in terms of its distance from the mean, measured with reference to standard deviation from the mean. Z-scores are useful in statistics because they allow comparison between two scores that belong to different normal distributions.
![**Problem Statement:**
Lisa has an average bowling score of 190 in her winter league where the average for all bowlers is 175 with a standard deviation of 9. Barb has an average bowling score of 175 in the spring league where the average for all bowlers is 155 with a standard deviation of 10.
(a) What is Lisa’s z-score? Round your answer to two decimal places as necessary.
(b) What is Barb’s z-score? Round your answer to two decimal places as necessary.
(c) Who scores higher in their own league if both league scores are normally distributed?
**Solution Explanation:**
To find the z-score, use the formula:
\[ z = \frac{X - \mu}{\sigma} \]
Where:
- \( z \) is the z-score,
- \( X \) is the individual score,
- \( \mu \) is the mean of the distribution,
- \( \sigma \) is the standard deviation of the distribution.
(a) **Lisa’s z-score:**
- \( X = 190 \)
- \( \mu = 175 \)
- \( \sigma = 9 \)
\[ z = \frac{190 - 175}{9} = \frac{15}{9} = 1.67 \]
(b) **Barb’s z-score:**
- \( X = 175 \)
- \( \mu = 155 \)
- \( \sigma = 10 \)
\[ z = \frac{175 - 155}{10} = \frac{20}{10} = 2.00 \]
(c) **Comparison:**
- Lisa’s z-score is 1.67.
- Barb’s z-score is 2.00.
Since Barb has a higher z-score, Barb scores higher relative to her league’s average than Lisa does in her league.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0ed02b96-00f1-4605-bbb0-113e667487ea%2F914fe66a-68f2-4f46-8ce8-6c53a7989195%2Fb8rxkk_processed.jpeg&w=3840&q=75)
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