Z-Scores and Standard Deviation

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Z-Scores and Standard Deviation · Grade 10 · Worksheet 2

1. A pharmaceutical company is testing a new medication. The average blood pressure reduction in clinical trials is 12 mmHg with a standard deviation of 3 mmHg. If a patient experiences a reduction of 18 mmHg, what is their z-score? Answer: 2 Solution: To find the z-score for a patient's blood pressure reduction, we use the z-score formula: z = (x - μ) / σ x = the individual data value (patient's blood pressure reduction) = 18 mmHg μ = the population mean (average reduction) = 12 mmHg σ = the population standard deviation = 3 mmHg z = (18 -…
   Full step-by-step solution

   To find the z-score for a patient's blood pressure reduction, we use the z-score formula: z = (x - μ) / σ Where: x = the individual data value (patient's blood pressure reduction) = 18 mmHg μ = the population mean (average reduction) = 12 mmHg σ = the population standard deviation = 3 mmHg Step 1: Substitute the known values into the formula z = (18 - 12) / 3 Step 2: Calculate the numerator (the difference from the mean) 18 - 12 = 6 So, z = 6 / 3 Step 3: Divide by the standard deviation 6 / 3 = 2 Therefore, the z-score is 2. Interpretation: A z-score of 2 means the patient's blood pressure reduction is 2 standard deviations above the average reduction.

2. A normal distribution of test scores has a mean of 75 and a standard deviation of 8. If a student scores 91 on the test, what is their z-score? Answer: 2 Solution: To find the z-score for a test score of 91 in a normal distribution with mean 75 and standard deviation 8, we use the z-score formula. Write down the z-score formula.
   Full step-by-step solution

   To find the z-score for a test score of 91 in a normal distribution with mean 75 and standard deviation 8, we use the z-score formula. Step 1: Write down the z-score formula. The z-score formula is: z = (x - mean) / standard deviation Step 2: Identify the given values from the problem. x = 91 (the student's score) mean = 75 standard deviation = 8 Step 3: Substitute the given values into the formula. z = (91 - 75) / 8 Step 4: Perform the subtraction inside the parentheses. 91 - 75 = 16 So, z = 16 / 8 Step 5: Perform the division. 16 divided by 8 equals 2. So, z = 2 Step 6: Interpret the result. A z-score of 2 means the student's score is 2 standard deviations above the mean. Therefore, the z-score is 2.

3. A normal distribution has μ = 82 and σ = 6. If x = 70, then z = ? Answer: -2 Solution: Recall the z-score formula: z = (x - μ) / σ Substitute the given values: z = (70 - 82) / 6 Calculate the numerator: 70 - 82 = -12 Divide by the standard deviation: -12 / 6 = -2 The z-score is -2, indicating the data point is 2 standard deviations below 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 = (70 - 82) / 6 Step 3: Calculate the numerator: 70 - 82 = -12 Step 4: Divide by the standard deviation: -12 / 6 = -2 Step 5: The z-score is -2, indicating the data point is 2 standard deviations below the mean. The answer is -2.

4. A normal distribution has μ = 82 and σ = 6. If x = 97, then z = ? Answer: 2.5 Solution: Recall the z-score formula: z = (x - μ) / σ Substitute the given values: z = (97 - 82) / 6 Calculate the numerator: 97 - 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: Recall the z-score formula: z = (x - μ) / σ Step 2: Substitute the given values: z = (97 - 82) / 6 Step 3: Calculate the numerator: 97 - 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.

5. A scientist measures the beak lengths of a population of finches on an island. The distribution of beak lengths is approximately normal with a mean of 16 mm and a standard deviation of 1 mm. One particular finch, observed by Sophia, has a beak length of 11 mm. Calculate the z-score for this finch's beak length and interpret what this value means in terms of the distribution. Answer: -5 Solution: Recall the z-score formula: z = (x - mu) / sigma Identify the values: x = 11 mm, mu = 16 mm, sigma = 1 mm Substitute into the formula: z = (11 - 16) / 1 Calculate the numerator: 11 - 16 = -5 Divide by the standard deviation: -5 / 1 = -5 Interpret the z-score: A z-score of -5 means that the…
   Full step-by-step solution

   Step 1: Recall the z-score formula: z = (x - mu) / sigma Step 2: Identify the values: x = 11 mm, mu = 16 mm, sigma = 1 mm Step 3: Substitute into the formula: z = (11 - 16) / 1 Step 4: Calculate the numerator: 11 - 16 = -5 Step 5: Divide by the standard deviation: -5 / 1 = -5 Step 6: Interpret the z-score: A z-score of -5 means that the finch's beak length is 5 standard deviations below the mean beak length of the population. The answer is -5.

6. A normal distribution of heights for 10th grade students has a mean of 165 cm with a standard deviation of 6 cm. If a student is 178 cm tall, what is their z-score? Answer: 2.17 Solution: Recall the z-score formula: z = (x - μ) / σ Identify the values: x = 178 cm, μ = 165 cm, σ = 6 cm Substitute into the formula: z = (178 - 165) / 6 Calculate the numerator: 178 - 165 = 13 Divide by the standard deviation: 13 / 6 = 2.1666...
   Full step-by-step solution

   Step 1: Recall the z-score formula: z = (x - μ) / σ Step 2: Identify the values: x = 178 cm, μ = 165 cm, σ = 6 cm Step 3: Substitute into the formula: z = (178 - 165) / 6 Step 4: Calculate the numerator: 178 - 165 = 13 Step 5: Divide by the standard deviation: 13 / 6 = 2.1666... Step 6: Round to two decimal places: 2.17 The z-score is 2.17.

7. 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.

8. A normal distribution of heart rates for athletes has a mean of 66 beats per minute and a standard deviation of 6 beats per minute. Sophia has a resting heart rate of 81 beats per minute. Calculate Sophia's z-score and interpret what it means in terms of standard deviations from the mean. Answer: 2.5 Solution: Recall the z-score formula: z = (x - mu) / sigma Identify the values: x = 81, mu = 66, sigma = 6 Substitute into the formula: z = (81 - 66) / 6 Calculate the numerator: 81 - 66 = 15 Divide by the standard deviation: 15 / 6 = 2.5 The z-score is 2.5.
   Full step-by-step solution

   Step 1: Recall the z-score formula: z = (x - mu) / sigma Step 2: Identify the values: x = 81, mu = 66, sigma = 6 Step 3: Substitute into the formula: z = (81 - 66) / 6 Step 4: Calculate the numerator: 81 - 66 = 15 Step 5: Divide by the standard deviation: 15 / 6 = 2.5 Step 6: The z-score is 2.5. This means Sophia's heart rate is 2.5 standard deviations above the mean heart rate of athletes. The answer is 2.5.