Z-Scores and Standard Deviation

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Answer Key & Explanations

Z-Scores and Standard Deviation · Grade 10 · Worksheet 1

1. A pharmaceutical company is testing a new medication and finds that the time it takes for patients to experience symptom relief follows a normal distribution with a mean of 45 minutes and a standard deviation of 8 minutes. If Liam takes this medication and experiences relief in 35 minutes, what is the z-score for his relief time? Answer: -1.25 Solution: Recall the formula for the z-score. The z-score tells us how many standard deviations a value is from the mean. z = (x - μ) / σ x = individual data value (Liam's relief time) μ = population mean σ = population standard deviation Identify the given values from the problem.
   Full step-by-step solution

   Step 1: Recall the formula for the z-score. The z-score tells us how many standard deviations a value is from the mean. The formula is: z = (x - μ) / σ where: x = individual data value (Liam's relief time) μ = population mean σ = population standard deviation Step 2: Identify the given values from the problem. Mean μ = 45 minutes Standard deviation σ = 8 minutes Liam's relief time x = 35 minutes Step 3: Substitute the values into the z-score formula. z = (35 - 45) / 8 Step 4: Perform the subtraction in the numerator. 35 - 45 = -10 So, z = (-10) / 8 Step 5: Divide the numerator by the denominator. -10 / 8 = -1.25 Step 6: Interpret the result. The z-score is -1.25, which means Liam's relief time is 1.25 standard deviations below the mean relief time. Final Answer: -1.25

2. Isabella is a quality control analyst at a juice bottling plant. The volume of juice in each bottle follows a normal distribution with a mean of 502 mL and a standard deviation of 3 mL. During a routine check, Isabella selects a bottle and finds it contains 511 mL of juice. Calculate the z-score for this bottle's volume and explain what it means. Answer: 3 Solution: Recall the z-score formula: z = (x - μ) / σ Identify the given values: x = 511 mL (the bottle's volume), μ = 502 mL (mean volume), σ = 3 mL (standard deviation) Substitute the values into the formula: z = (511 - 502) / 3 Calculate the numerator: 511 - 502 = 9 Divide by the standard deviation: 9…
   Full step-by-step solution

   Step 1: Recall the z-score formula: z = (x - μ) / σ Step 2: Identify the given values: x = 511 mL (the bottle's volume), μ = 502 mL (mean volume), σ = 3 mL (standard deviation) Step 3: Substitute the values into the formula: z = (511 - 502) / 3 Step 4: Calculate the numerator: 511 - 502 = 9 Step 5: Divide by the standard deviation: 9 / 3 = 3 Step 6: The z-score is 3, which means this bottle's volume is 3 standard deviations above the mean. The answer is 3.

3. Matiu is a botanist studying the growth of a rare tree species in a controlled environment. The heights of these trees after two years follow a normal distribution with a mean of 45 cm and a standard deviation of 8 cm. Matiu measures one particular tree and finds it to be 61 cm tall. What is the z-score for this tree's height, and what does it indicate? Answer: 2 Solution: Recall the z-score formula: z = (x - μ) / σ, where x is the data point, μ is the mean, and σ is the standard deviation. Identify the values from the problem: x = 61 cm, μ = 45 cm, σ = 8 cm.
   Full step-by-step solution

   Step 1: Recall the z-score formula: z = (x - μ) / σ, where x is the data point, μ is the mean, and σ is the standard deviation. Step 2: Identify the values from the problem: x = 61 cm, μ = 45 cm, σ = 8 cm. Step 3: Substitute the values into the formula: z = (61 - 45) / 8. Step 4: Calculate the numerator: 61 - 45 = 16. Step 5: Divide by the standard deviation: 16 / 8 = 2. Step 6: The z-score is 2, which means this tree's height is 2 standard deviations above the mean height of the species. The answer is 2.

4. A graph shows the distribution of marathon finish times for 10th-grade runners. The distribution is approximately normal with a mean of 245 minutes and a standard deviation of 18 minutes. Hana completes the marathon in 281 minutes. Calculate Hana's z-score and explain what it means in terms of standard deviations from the mean. Answer: 2 Solution: Recall the z-score formula: z = (x - μ) / σ Identify the values: x = 281 minutes (Hana's time), μ = 245 minutes (mean), σ = 18 minutes (standard deviation) Substitute into the formula: z = (281 - 245) / 18 Calculate the numerator: 281 - 245 = 36 Divide by the standard deviation: 36 / 18 = 2…
   Full step-by-step solution

   Step 1: Recall the z-score formula: z = (x - μ) / σ Step 2: Identify the values: x = 281 minutes (Hana's time), μ = 245 minutes (mean), σ = 18 minutes (standard deviation) Step 3: Substitute into the formula: z = (281 - 245) / 18 Step 4: Calculate the numerator: 281 - 245 = 36 Step 5: Divide by the standard deviation: 36 / 18 = 2 Step 6: Interpret the z-score: A z-score of 2 means Hana's finish time is 2 standard deviations above the mean. The answer is 2.

5. The average score on a national mathematics exam is 78 points with a standard deviation of 5 points. If Sarah scored 86 points on this exam, what is her z-score? Round your answer to two decimal places. Answer: 1.60 Solution: Recall the z-score formula. The z-score measures how many standard deviations a data point is from the mean. z = (x - μ) / σ x = individual score μ = population mean σ = population standard deviation Identify the given values from the problem.
   Full step-by-step solution

   Step 1: Recall the z-score formula. The z-score measures how many standard deviations a data point is from the mean. The formula is: z = (x - μ) / σ where: x = individual score μ = population mean σ = population standard deviation Step 2: Identify the given values from the problem. Mean (μ) = 78 Standard deviation (σ) = 5 Sarah’s score (x) = 86 Step 3: Substitute the values into the formula. z = (86 - 78) / 5 Step 4: Perform the subtraction in the numerator. 86 - 78 = 8 So, z = 8 / 5 Step 5: Divide 8 by 5. 8 / 5 = 1.6 Step 6: Round to two decimal places. 1.6 is already 1.60 when written to two decimal places. Step 7: Interpret the result. Sarah’s z-score is 1.60, meaning her score is 1.60 standard deviations above the mean. Final Answer: 1.60

6. Noah is monitoring the water quality of a local river. The pH levels of the river water follow a normal distribution with a mean of 7.4 and a standard deviation of 0.3. During a routine check, Noah measures a pH level of 6.8 at a specific location. What is the z-score for this pH measurement? Answer: -2 Solution: Recall the z-score formula: z = (x - μ) / σ Identify the values from the problem: x = 6.8, μ = 7.4, σ = 0.3 Substitute the values into the formula: z = (6.8 - 7.4) / 0.3 Calculate the numerator: 6.8 - 7.4 = -0.6 Divide by the standard deviation: -0.6 / 0.3 = -2 The z-score is -2, meaning this pH…
   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 = 6.8, μ = 7.4, σ = 0.3 Step 3: Substitute the values into the formula: z = (6.8 - 7.4) / 0.3 Step 4: Calculate the numerator: 6.8 - 7.4 = -0.6 Step 5: Divide by the standard deviation: -0.6 / 0.3 = -2 Step 6: The z-score is -2, meaning this pH measurement is 2 standard deviations below the mean. The answer is -2.

7. A normally distributed dataset has μ = 82 and σ = 5. Find the z-score for x = 94. Answer: 2.4 Solution: Recall the z-score formula: z = (x - μ) / σ Substitute the given values: z = (94 - 82) / 5 Calculate the numerator: 94 - 82 = 12 Divide by the standard deviation: 12 ÷ 5 = 2.4 The z-score is 2.4, meaning the data point is 2.4 standard deviations above the mean.
   Full step-by-step solution

   Step 1: Recall the z-score formula: z = (x - μ) / σ Step 2: Substitute the given values: z = (94 - 82) / 5 Step 3: Calculate the numerator: 94 - 82 = 12 Step 4: Divide by the standard deviation: 12 ÷ 5 = 2.4 Step 5: The z-score is 2.4, meaning the data point is 2.4 standard deviations above the mean.