Q1: A z-score tells you how many __________ a data value is from the mean. (a) units (b) standard deviations (c) percentiles (d) quartiles
Solution:
Ans: (b) Explanation: A z-score measures the number of standard deviations a particular data value is away from the mean. It standardizes values so they can be compared across different distributions.
Q2: What is the z-score of the mean in any normal distribution? (a) 1 (b) 0 (c) -1 (d) Cannot be determined
Solution:
Ans: (b) Explanation: The mean has a z-score of 0 because it is 0 standard deviations away from itself. Using the formula \(z = \frac{x - \mu}{\sigma}\), when \(x = \mu\), we get \(z = 0\).
Q3: If a data value has a z-score of -2.5, what does this indicate? (a) The value is 2.5 standard deviations above the mean (b) The value is 2.5 standard deviations below the mean (c) The value is equal to the mean (d) The value is 2.5 units below the mean
Solution:
Ans: (b) Explanation: A negative z-score indicates the data value is below the mean. The value -2.5 means the data point is 2.5 standard deviations below the mean. Option (a) is incorrect because positive z-scores indicate values above the mean, and option (d) confuses units with standard deviations.
Q4: Which formula correctly represents the calculation of a z-score? (a) \(z = \frac{\mu - x}{\sigma}\) (b) \(z = \frac{x - \mu}{\sigma}\) (c) \(z = \frac{\sigma}{x - \mu}\) (d) \(z = \frac{x + \mu}{\sigma}\)
Solution:
Ans: (b) Explanation: The correct z-score formula is \(z = \frac{x - \mu}{\sigma}\), where \(x\) is the data value, \(\mu\) is the mean, and \(\sigma\) is the standard deviation. This formula measures how many standard deviations \(x\) is from the mean.
Q5: A student scored 85 on a test where the mean was 75 and the standard deviation was 5. What is the student's z-score? (a) 2 (b) 10 (c) -2 (d) 1.7
Solution:
Ans: (a) Explanation: Using the formula \(z = \frac{x - \mu}{\sigma}\): \(z = \frac{85 - 75}{5} = \frac{10}{5} = 2\) The student's score is 2 standard deviations above the mean. Option (b) represents the raw difference, not the z-score.
Q6: If a z-score is 0, which statement is true? (a) The value is the highest in the dataset (b) The value equals the mean (c) The value is one standard deviation above the mean (d) The value is below the mean
Solution:
Ans: (b) Explanation: A z-score of 0 means the data value is exactly at the mean. When \(x = \mu\), the numerator \(x - \mu = 0\), making \(z = 0\).
Q7: Which z-score represents a value that is most unusual or extreme in a normal distribution? (a) 0.5 (b) 1.2 (c) -3.8 (d) -0.3
Solution:
Ans: (c) Explanation: The larger the absolute value of a z-score, the more unusual or extreme the data value is. A z-score of -3.8 is the farthest from the mean among the options, making it the most extreme. Values beyond ±3 standard deviations are considered very rare in a normal distribution.
Q8: In a normal distribution, approximately what percentage of data falls within 1 standard deviation of the mean (between z = -1 and z = 1)? (a) 50% (b) 68% (c) 95% (d) 99.7%
Solution:
Ans: (b) Explanation: According to the Empirical Rule (68-95-99.7 rule), approximately 68% of data in a normal distribution falls within one standard deviation of the mean. This is a fundamental property of normal distributions.
Section B: Fill in the Blanks
Q9: The z-score formula is written as \(z = \frac{x - \mu}{__________}\).
Solution:
Ans: \(\sigma\) or standard deviation Explanation: The denominator of the z-score formula is the standard deviation (represented by \(\sigma\)), which standardizes the deviation from the mean.
Q10: A positive z-score indicates that the data value is __________ the mean.
Solution:
Ans: above or greater than Explanation: When the data value \(x\) is greater than the mean \(\mu\), the numerator \(x - \mu\) is positive, resulting in a positive z-score.
Q11: In the Empirical Rule, approximately 95% of data falls within __________ standard deviations of the mean.
Solution:
Ans: 2 or two Explanation: The Empirical Rule states that approximately 95% of data in a normal distribution falls within 2 standard deviations of the mean (between z = -2 and z = 2).
Q12: Z-scores allow us to compare values from different distributions by __________ the data.
Solution:
Ans: standardizing Explanation:Standardizing data using z-scores converts values to a common scale with mean 0 and standard deviation 1, allowing for meaningful comparisons across different distributions.
Q13: If a z-score is negative, the original data value is __________ the mean.
Solution:
Ans: below or less than Explanation: A negative z-score results when the data value \(x\) is less than the mean \(\mu\), making the numerator \(x - \mu\) negative.
Q14: The process of converting a raw score to a z-score is called __________.
Solution:
Ans: standardization or standardizing Explanation:Standardization is the process of transforming raw data into z-scores, creating a distribution with mean 0 and standard deviation 1.
Section C: Word Problems
Q15: The heights of students in a class are normally distributed with a mean of 165 cm and a standard deviation of 8 cm. Calculate the z-score for a student who is 181 cm tall.
Solution:
Ans: Given: \(\mu = 165\) cm, \(\sigma = 8\) cm, \(x = 181\) cm Using the formula: \(z = \frac{x - \mu}{\sigma}\) \(z = \frac{181 - 165}{8} = \frac{16}{8} = 2\) Final Answer: The z-score is 2
Q16: On a standardized test, the mean score is 500 and the standard deviation is 100. If a student scored 650, what is their z-score? Interpret what this z-score means.
Solution:
Ans: Given: \(\mu = 500\), \(\sigma = 100\), \(x = 650\) Using the formula: \(z = \frac{x - \mu}{\sigma}\) \(z = \frac{650 - 500}{100} = \frac{150}{100} = 1.5\) Final Answer: The z-score is 1.5, which means the student scored 1.5 standard deviations above the mean
Q17: The average time to complete a marathon is 4.5 hours with a standard deviation of 0.5 hours. A runner completed the marathon in 3.75 hours. Calculate the z-score for this runner's time.
Solution:
Ans: Given: \(\mu = 4.5\) hours, \(\sigma = 0.5\) hours, \(x = 3.75\) hours Using the formula: \(z = \frac{x - \mu}{\sigma}\) \(z = \frac{3.75 - 4.5}{0.5} = \frac{-0.75}{0.5} = -1.5\) Final Answer: The z-score is -1.5
Q18: In a chemistry class, the mean test score is 78 with a standard deviation of 6. Maria's z-score on the test was -2. What was Maria's actual test score?
Solution:
Ans: Given: \(\mu = 78\), \(\sigma = 6\), \(z = -2\) Using the formula: \(z = \frac{x - \mu}{\sigma}\), we solve for \(x\): \(-2 = \frac{x - 78}{6}\) \(-12 = x - 78\) \(x = 78 - 12 = 66\) Final Answer: Maria's test score was 66
Q19: The weights of apples in an orchard are normally distributed with a mean of 150 grams and a standard deviation of 15 grams. An apple weighing 120 grams is selected. Find the z-score and determine if this apple is considered unusually small (z-score less than -2).
Solution:
Ans: Given: \(\mu = 150\) grams, \(\sigma = 15\) grams, \(x = 120\) grams Using the formula: \(z = \frac{x - \mu}{\sigma}\) \(z = \frac{120 - 150}{15} = \frac{-30}{15} = -2\) Final Answer: The z-score is -2. Since z = -2, this apple is at the boundary and would be considered unusually small
Q20: Two students took different mathematics exams. Student A scored 82 on an exam with mean 70 and standard deviation 8. Student B scored 90 on an exam with mean 80 and standard deviation 5. Using z-scores, determine which student performed better relative to their classmates.
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