Z-Scores and Standard Deviation

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Answer Key & Explanations

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

1. A normal distribution has μ = 82 and σ = 6. If x = 94, then z = ? Answer: 2 Solution: Recall the z-score formula: z = (x - μ) / σ Substitute the given values: z = (94 - 82) / 6 Calculate the numerator: 94 - 82 = 12 Divide by the standard deviation: 12 / 6 = 2 The z-score is 2, meaning 94 is 2 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) / 6 Step 3: Calculate the numerator: 94 - 82 = 12 Step 4: Divide by the standard deviation: 12 / 6 = 2 Step 5: The z-score is 2, meaning 94 is 2 standard deviations above the mean. The answer is 2.

2. z = (82 - 70) ÷ 4 = ? Answer: 3 Solution: Calculate the difference between the data value and the mean: 82 - 70 = 12 Divide this difference by the standard deviation: 12 ÷ 4 = 3 The result is the z-score: z = 3 The answer is 3.
   Full step-by-step solution

   Step 1: Calculate the difference between the data value and the mean: 82 - 70 = 12 Step 2: Divide this difference by the standard deviation: 12 ÷ 4 = 3 Step 3: The result is the z-score: z = 3 The answer is 3.

3. A botanist measures the petal lengths of a rare flower species. The distribution of petal lengths is normal with a mean of 7.2 cm and a standard deviation of 1.7 cm. One particular flower, collected by Charlotte, has a petal length of 10.6 cm. Draw a normal distribution curve in your mind, place the mean at the center, and mark Charlotte's flower on the curve. What is the z-score for this flower's petal length, and what does it tell you about how its length compares to the rest of the population? Answer: 2 Solution: Write the z-score formula: z = (x - mu) / sigma. Step 2: Identify the values: x = 10.6 cm, mu = 7.2 cm, sigma = 1.7 cm. Step 3: Substitute into the formula: z = (10.6 - 7.2) / 1.7.
   Full step-by-step solution

   Step 1: Write the z-score formula: z = (x - mu) / sigma. Step 2: Identify the values: x = 10.6 cm, mu = 7.2 cm, sigma = 1.7 cm. Step 3: Substitute into the formula: z = (10.6 - 7.2) / 1.7. Step 4: Calculate the numerator: 10.6 - 7.2 = 3.4. Step 5: Divide by the standard deviation: 3.4 / 1.7 = 2. Step 6: Interpret: A z-score of 2 means Charlotte's flower petal length is 2 standard deviations above the mean, so it is longer than most flowers in the population. The answer is 2.

4. Charlotte is a meteorologist studying rainfall in her city. The annual rainfall follows a normal distribution with a mean of 72 cm and a standard deviation of 12 cm. Last year, the city recorded 87 cm of rainfall. What is the z-score for last year's rainfall, and what does it indicate about the rainfall compared to the average? Answer: 1.25 Solution: Write the z-score formula: z = (x - μ) / σ Identify the values: x = 87 cm (last year's rainfall), μ = 72 cm (mean rainfall), σ = 12 cm (standard deviation) Substitute into the formula: z = (87 - 72) / 12 Calculate the numerator: 87 - 72 = 15 Divide by the standard deviation: 15 / 12 = 1.25…
   Full step-by-step solution

   Step 1: Write the z-score formula: z = (x - μ) / σ Step 2: Identify the values: x = 87 cm (last year's rainfall), μ = 72 cm (mean rainfall), σ = 12 cm (standard deviation) Step 3: Substitute into the formula: z = (87 - 72) / 12 Step 4: Calculate the numerator: 87 - 72 = 15 Step 5: Divide by the standard deviation: 15 / 12 = 1.25 Step 6: Interpret the z-score: A z-score of 1.25 means last year's rainfall was 1.25 standard deviations above the mean annual rainfall. The answer is 1.25.

5. Isabella is studying the fuel efficiency of a new hybrid car model. The fuel efficiency (in miles per gallon) for this car model follows a normal distribution with a mean of 52 mpg and a standard deviation of 4 mpg. During a test drive, Isabella records that a particular car achieves 44 mpg. Calculate the z-score for this fuel efficiency measurement and interpret its meaning. 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 = 44 mpg, μ = 52 mpg, σ = 4 mpg.
   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 = 44 mpg, μ = 52 mpg, σ = 4 mpg. Step 3: Substitute the values into the formula: z = (44 - 52) / 4. Step 4: Calculate the numerator: 44 - 52 = -8. Step 5: Divide by the standard deviation: -8 / 4 = -2. Step 6: The z-score is -2. This means that the car's fuel efficiency of 44 mpg is 2 standard deviations below the mean fuel efficiency of 52 mpg. In other words, it is significantly lower than average. The answer is -2.

6. A normal distribution has μ = 82 and σ = 6. If x = 67, then z = ? Answer: -2.5 Solution: Write the z-score formula: z = (x - μ) / σ Substitute the given values: z = (67 - 82) / 6 Calculate the numerator: 67 - 82 = -15 Divide by the standard deviation: -15 / 6 = -2.5 The z-score is -2.5 The answer is -2.5.
   Full step-by-step solution

   Step 1: Write the z-score formula: z = (x - μ) / σ Step 2: Substitute the given values: z = (67 - 82) / 6 Step 3: Calculate the numerator: 67 - 82 = -15 Step 4: Divide by the standard deviation: -15 / 6 = -2.5 Step 5: The z-score is -2.5 The answer is -2.5.

7. A histogram shows the distribution of marathon finish times for 10th grade runners. The mean finish time is 246 minutes with a standard deviation of 21 minutes. Sophia finishes the marathon in 281 minutes. Calculate Sophia's z-score and interpret what it means in terms of standard deviations from the mean. Answer: 1.67 Solution: Recall the z-score formula: z = (x - μ) / σ Identify the values: x = 281 minutes (Sophia's time), μ = 246 minutes (mean), σ = 21 minutes (standard deviation) Substitute into the formula: z = (281 - 246) / 21 Calculate the numerator: 281 - 246 = 35 Divide by the standard deviation: 35 / 21 =…
   Full step-by-step solution

   Step 1: Recall the z-score formula: z = (x - μ) / σ Step 2: Identify the values: x = 281 minutes (Sophia's time), μ = 246 minutes (mean), σ = 21 minutes (standard deviation) Step 3: Substitute into the formula: z = (281 - 246) / 21 Step 4: Calculate the numerator: 281 - 246 = 35 Step 5: Divide by the standard deviation: 35 / 21 = 1.6666... Step 6: Round to two decimal places: 1.67 Step 7: Interpretation: Sophia's finish time is 1.67 standard deviations above the mean. The answer is 1.67.