Z-Scores and Standard Deviation

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Z-Scores and Standard Deviation · Grade 10 · Worksheet 3

1. A university is studying the heights of its basketball players. The team's heights follow a normal distribution with a mean of 192 cm and a standard deviation of 6 cm. If Noah, a player on the team, has a height of 204 cm, what is his z-score? Answer: 2 Solution: Recall the z-score formula: z = (x - μ) / σ Identify the values: x = 204 cm, μ = 192 cm, σ = 6 cm Substitute into the formula: z = (204 - 192) / 6 Calculate the numerator: 204 - 192 = 12 Divide by the standard deviation: 12 / 6 = 2 The z-score is 2, meaning Noah's height is 2 standard deviations…
   Full step-by-step solution

   Step 1: Recall the z-score formula: z = (x - μ) / σ Step 2: Identify the values: x = 204 cm, μ = 192 cm, σ = 6 cm Step 3: Substitute into the formula: z = (204 - 192) / 6 Step 4: Calculate the numerator: 204 - 192 = 12 Step 5: Divide by the standard deviation: 12 / 6 = 2 Step 6: The z-score is 2, meaning Noah's height is 2 standard deviations above the mean.

2. The heights of students in a Grade 10 class follow a normal distribution with a mean of 165 cm and a standard deviation of 6 cm. If a student is 178 cm tall, what is their z-score? Round your answer to two decimal places. Answer: 2.17 Solution: Mean (μ) = 165 cm Standard deviation (σ) = 6 cm Student's height (x) = 178 cm z = (x - μ) / σ z = (178 - 165) / 6 178 - 165 = 13 z = 13 / 6 13 ÷ 6 = 2.1666... 2.1666...
   Full step-by-step solution

   Step 1: Identify the given values Mean (μ) = 165 cm Standard deviation (σ) = 6 cm Student's height (x) = 178 cm Step 2: Write the z-score formula z = (x - μ) / σ Step 3: Substitute the values into the formula z = (178 - 165) / 6 Step 4: Calculate the numerator 178 - 165 = 13 Step 5: Divide by the standard deviation z = 13 / 6 Step 6: Calculate the division 13 ÷ 6 = 2.1666... Step 7: Round to two decimal places 2.1666... rounded to two decimal places is 2.17 The answer is 2.17.

3. A box-and-whisker plot shows the distribution of test scores for a Grade 10 class. The mean score is 73 and the standard deviation is 9. Aroha's score is represented on the plot, and her z-score is 3. What is Aroha's actual test score? Answer: 100 Solution: Recall the z-score formula: z = (x - μ) / σ, where x is the score, μ is the mean, and σ is the standard deviation. Substitute the given values: z = 3, μ = 73, σ = 9. Set up the equation: 3 = (x - 73) / 9.
   Full step-by-step solution

   Step 1: Recall the z-score formula: z = (x - μ) / σ, where x is the score, μ is the mean, and σ is the standard deviation. Step 2: Substitute the given values: z = 3, μ = 73, σ = 9. Step 3: Set up the equation: 3 = (x - 73) / 9. Step 4: Multiply both sides by 9: 3 * 9 = x - 73, so 27 = x - 73. Step 5: Add 73 to both sides: x = 27 + 73 = 100. Step 6: Aroha's actual test score is 100. The answer is 100.

4. A dot plot shows the test scores for a Grade 10 class. The distribution is roughly symmetric and bell-shaped, with a mean score of 78 and a standard deviation of 9. Noah scores a 96 on the test. Calculate Noah's z-score and interpret what it means in terms of the distribution. Answer: 2 Solution: Recall the z-score formula: z = (x - mu) / sigma Identify the values: x = 96 (Noah's score), mu = 78 (mean), sigma = 9 (standard deviation) Substitute into the formula: z = (96 - 78) / 9 Calculate the numerator: 96 - 78 = 18 Divide by the standard deviation: 18 / 9 = 2 Interpret the result: A…
   Full step-by-step solution

   Step 1: Recall the z-score formula: z = (x - mu) / sigma Step 2: Identify the values: x = 96 (Noah's score), mu = 78 (mean), sigma = 9 (standard deviation) Step 3: Substitute into the formula: z = (96 - 78) / 9 Step 4: Calculate the numerator: 96 - 78 = 18 Step 5: Divide by the standard deviation: 18 / 9 = 2 Step 6: Interpret the result: A z-score of 2 means Noah's score is 2 standard deviations above the mean of the class. The answer is 2.

5. Hana is studying the growth of kauri trees in a protected forest. The heights of mature kauri trees in this forest follow a normal distribution with a mean height of 42 meters and a standard deviation of 4 meters. Hana measures one particularly tall kauri tree and finds its height is 54 meters. What is the z-score for this tree's height, and what does it indicate about how this tree compares to the average kauri tree? Answer: 3 Solution: Recall the z-score formula: z = (x - mu) / sigma Identify the values from the problem: x = 54 meters, mu = 42 meters, sigma = 4 meters Substitute the values into the formula: z = (54 - 42) / 4 Calculate the numerator: 54 - 42 = 12 Divide by the standard deviation: 12 / 4 = 3 The z-score is 3.
   Full step-by-step solution

   Step 1: Recall the z-score formula: z = (x - mu) / sigma Step 2: Identify the values from the problem: x = 54 meters, mu = 42 meters, sigma = 4 meters Step 3: Substitute the values into the formula: z = (54 - 42) / 4 Step 4: Calculate the numerator: 54 - 42 = 12 Step 5: Divide by the standard deviation: 12 / 4 = 3 Step 6: The z-score is 3. This means the height of this kauri tree is 3 standard deviations above the mean height of all mature kauri trees in the forest. The answer is 3.

6. Mason is studying the time it takes for a particular chemical reaction to reach completion under controlled conditions. The reaction times follow a normal distribution with a mean of 47 seconds and a standard deviation of 2 seconds. If one trial of the reaction completes in 52 seconds, what is the z-score for this reaction time? Interpret what this z-score means. Answer: 2.5 Solution: Recall the z-score formula: z = (x - μ) / σ Identify the values from the problem: x = 52 seconds, μ = 47 seconds, σ = 2 seconds Substitute the values into the formula: z = (52 - 47) / 2 Calculate the numerator: 52 - 47 = 5 Divide by the standard deviation: 5 / 2 = 2.5 The z-score is 2.5, which…
   Full step-by-step solution

   Step 1: Recall the z-score formula: z = (x - μ) / σ Step 2: Identify the values from the problem: x = 52 seconds, μ = 47 seconds, σ = 2 seconds Step 3: Substitute the values into the formula: z = (52 - 47) / 2 Step 4: Calculate the numerator: 52 - 47 = 5 Step 5: Divide by the standard deviation: 5 / 2 = 2.5 Step 6: The z-score is 2.5, which means this reaction time is 2.5 standard deviations above the mean. This indicates that the reaction took significantly longer than the average time. The answer is 2.5.

7. Kaia is analyzing the germination times of a rare native plant species. The germination times (in days) follow a normal distribution with a mean of 21 days and a standard deviation of 3 days. One particular seed from Kaia's experiment germinated in 15 days. Calculate the z-score for this germination time and interpret what this value means in terms of standard deviations from the mean. Answer: -2 Solution: Recall the z-score formula: z = (x - μ) / σ Identify the values from the problem: x = 15 days, μ = 21 days, σ = 3 days Substitute the values into the formula: z = (15 - 21) / 3 Calculate the numerator: 15 - 21 = -6 Divide by the standard deviation: -6 / 3 = -2 The z-score is -2, which means this…
   Full step-by-step solution

   Step 1: Recall the z-score formula: z = (x - μ) / σ Step 2: Identify the values from the problem: x = 15 days, μ = 21 days, σ = 3 days Step 3: Substitute the values into the formula: z = (15 - 21) / 3 Step 4: Calculate the numerator: 15 - 21 = -6 Step 5: Divide by the standard deviation: -6 / 3 = -2 Step 6: The z-score is -2, which means this germination time is 2 standard deviations below the mean. The answer is -2.