Q1: A normal distribution is symmetric about which measure of center? (a) Range (b) Mean (c) Interquartile range (d) Variance
Solution:
Ans: (b) Explanation: A normal distribution is a bell-shaped curve that is perfectly symmetric about its mean. The mean is located at the center of the distribution, and the curve has equal areas on both sides of it.
Q2: In a normal distribution, approximately what percentage of data falls within one standard deviation of the mean? (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 the data in a normal distribution falls within one standard deviation of the mean. This is \(\mu \pm \sigma\).
Q3: A z-score of 0 indicates that a data value is: (a) One standard deviation above the mean (b) Equal to the mean (c) One standard deviation below the mean (d) Two standard deviations above the mean
Solution:
Ans: (b) Explanation: The z-score measures how many standard deviations a value is from the mean. A z-score of 0 means the data value is exactly equal to the mean, using the formula \(z = \frac{x - \mu}{\sigma}\).
Q4: If a normal distribution has a mean of 100 and a standard deviation of 15, what is the z-score for a value of 115? (a) 0 (b) 1 (c) -1 (d) 15
Solution:
Ans: (b) Explanation: Using the formula \(z = \frac{x - \mu}{\sigma}\), we calculate: \(z = \frac{115 - 100}{15} = \frac{15}{15} = 1\). The value 115 is one standard deviation above the mean, giving a z-score of 1.
Q5: According to the Empirical Rule, approximately what percentage of data in a normal distribution falls within two standard deviations of the mean? (a) 68% (b) 75% (c) 95% (d) 99.7%
Solution:
Ans: (c) Explanation: The Empirical Rule states that approximately 95% of data in a normal distribution falls within two standard deviations of the mean, represented by \(\mu \pm 2\sigma\).
Q6: Which of the following z-scores indicates a value furthest from the mean? (a) z = -0.5 (b) z = 1.2 (c) z = -2.3 (d) z = 0.8
Solution:
Ans: (c) Explanation: The distance from the mean is measured by the absolute value of the z-score. Among the options, \(|-2.3| = 2.3\) is the largest, meaning z = -2.3 represents a value furthest from the mean.
Q7: In a normal distribution, the mean is 50 and the standard deviation is 10. What percentage of data is expected to fall between 30 and 70? (a) 50% (b) 68% (c) 95% (d) 99.7%
Solution:
Ans: (c) Explanation: The values 30 and 70 are calculated as follows: 30 = 50 - 2(10) and 70 = 50 + 2(10) This represents \(\mu \pm 2\sigma\), which according to the Empirical Rule contains approximately 95% of the data.
Q8: A standardized test has scores that are normally distributed with a mean of 500 and a standard deviation of 100. A student who scores 400 has a z-score of: (a) -1 (b) 1 (c) -100 (d) 0.5
Solution:
Ans: (a) Explanation: Using the z-score formula \(z = \frac{x - \mu}{\sigma}\): \(z = \frac{400 - 500}{100} = \frac{-100}{100} = -1\) The score of 400 is one standard deviation below the mean, resulting in a z-score of -1.
Section B: Fill in the Blanks
Q9: A normal distribution is often called a __________ curve due to its characteristic shape.
Solution:
Ans: bell Explanation: The normal distribution has a symmetric, bell-shaped appearance, which is why it is commonly referred to as a bell curve.
Q10: The formula for calculating a z-score is z = __________, where x is the data value, μ is the mean, and σ is the standard deviation.
Solution:
Ans: \(\frac{x - \mu}{\sigma}\) Explanation: The z-score formula \(z = \frac{x - \mu}{\sigma}\) standardizes a data value by expressing how many standard deviations it is from the mean.
Q11: In a normal distribution, the mean, median, and mode are all __________ to each other.
Solution:
Ans: equal Explanation: Due to the symmetry of a normal distribution, the mean, median, and mode all have the same value and are located at the center of the distribution.
Q12: According to the Empirical Rule, approximately 99.7% of data in a normal distribution falls within __________ standard deviations of the mean.
Solution:
Ans: three Explanation: The Empirical Rule states that approximately 99.7% of data falls within three standard deviations (\(\mu \pm 3\sigma\)) of the mean in a normal distribution.
Q13: A z-score that is negative indicates that the data value is __________ the mean.
Solution:
Ans: below Explanation: A negative z-score means the data value is less than the mean, or below it. A positive z-score indicates the value is above the mean.
Q14: The total area under a normal distribution curve equals __________.
Solution:
Ans: 1 (or 100%) Explanation: The total area under the normal curve represents all possible outcomes and equals 1 (or 100% when expressed as a percentage), representing the entire probability space.
Section C: Word Problems
Q15: The heights of students in a school are normally distributed with a mean of 65 inches and a standard deviation of 3 inches. What percentage of students have heights between 62 and 68 inches?
Solution:
Ans: Step 1: Identify the range in terms of standard deviations. Lower bound: \(62 = 65 - 3 = \mu - \sigma\) Upper bound: \(68 = 65 + 3 = \mu + \sigma\)
Step 2: Apply the Empirical Rule. The range \(\mu \pm \sigma\) contains approximately 68% of the data.
Final Answer: Approximately 68% of students have heights between 62 and 68 inches.
Q16: Test scores in a large class are normally distributed with a mean of 75 and a standard deviation of 8. Calculate the z-score for a student who scored 91 on the test.
Solution:
Ans: Step 1: Use the z-score formula: \(z = \frac{x - \mu}{\sigma}\)
Q17: The weights of apples in an orchard are normally distributed with a mean of 150 grams and a standard deviation of 20 grams. Between what two weights do approximately 95% of the apples fall?
Solution:
Ans: Step 1: Apply the Empirical Rule. 95% of data falls within \(\mu \pm 2\sigma\).
Final Answer: Approximately 95% of apples weigh between 110 grams and 190 grams.
Q18: A factory produces bolts with lengths that are normally distributed. The mean length is 5 cm with a standard deviation of 0.2 cm. A bolt measuring 4.4 cm is found. Calculate the z-score for this bolt and determine if it is unusual (a z-score beyond ±2 is typically considered unusual).
Step 2: Interpret the z-score. Since \(|-3| = 3 > 2\), the bolt is more than 2 standard deviations from the mean.
Final Answer: The z-score is -3, and this bolt is considered unusual.
Q19: The times it takes for students to complete a standardized test are normally distributed with a mean of 90 minutes and a standard deviation of 12 minutes. What time represents a z-score of 1.5?
Solution:
Ans: Step 1: Use the z-score formula and solve for x: \(z = \frac{x - \mu}{\sigma}\) Rearranging: \(x = \mu + z\sigma\)
Q20: In a city, the daily high temperatures in July are normally distributed with a mean of 85°F and a standard deviation of 5°F. On what percentage of days would you expect the temperature to be between 75°F and 95°F?
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