Z-Scores and Standard Deviation

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

1. Charlotte is studying the battery life of a new smartphone model. The battery life follows a normal distribution with a mean of 18 hours and a standard deviation of 3 hours. If one smartphone from the production line has a battery life of 24 hours, what is the z-score for this smartphone's battery life, and what does it indicate about its performance relative to the average? Answer: 2 Solution: Recall the z-score formula: z = (x - μ) / σ Identify the values: x = 24 hours (the battery life of the smartphone), μ = 18 hours (the mean battery life), σ = 3 hours (the standard deviation) Substitute into the formula: z = (24 - 18) / 3 Calculate the numerator: 24 - 18 = 6 Divide by the…
   Full step-by-step solution

   Step 1: Recall the z-score formula: z = (x - μ) / σ Step 2: Identify the values: x = 24 hours (the battery life of the smartphone), μ = 18 hours (the mean battery life), σ = 3 hours (the standard deviation) Step 3: Substitute into the formula: z = (24 - 18) / 3 Step 4: Calculate the numerator: 24 - 18 = 6 Step 5: Divide by the standard deviation: 6 / 3 = 2 Step 6: The z-score is 2. This means the smartphone's battery life is 2 standard deviations above the mean, indicating it performs significantly better than the average smartphone. The answer is 2.

2. Olivia is a botanist studying the germination times of a rare orchid species. The germination times follow a normal distribution with a mean of 25 days and a standard deviation of 5 days. One particular seed she is observing germinates in 35 days. What is the z-score for this germination time, and what does it indicate about how this seed compares to the average? Answer: 2 Solution: Recall the z-score formula: z = (x - μ) / σ Identify the values from the problem: x = 35 days (the seed's germination time), μ = 25 days (mean germination time), σ = 5 days (standard deviation) Substitute the values into the formula: z = (35 - 25) / 5 Calculate the numerator: 35 - 25 = 10 Divide…
   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 = 35 days (the seed's germination time), μ = 25 days (mean germination time), σ = 5 days (standard deviation) Step 3: Substitute the values into the formula: z = (35 - 25) / 5 Step 4: Calculate the numerator: 35 - 25 = 10 Step 5: Divide by the standard deviation: 10 / 5 = 2 Step 6: The z-score is 2. This means the seed's germination time is 2 standard deviations above the mean, indicating it took significantly longer to germinate than the average seed. The answer is 2.

3. Sophia is training for a regional cross-country meet and has been tracking her running times. Her coach tells her that the times for all runners in her age group follow a normal distribution with a mean time of 28 minutes and a standard deviation of 4 minutes. If Sophia's personal best time is 22 minutes, what is her z-score and what does it indicate about her performance relative to other runners? Answer: -1.5 Solution: Recall the z-score formula: z = (x - μ) / σ, where x is the data value, μ is the mean, and σ is the standard deviation.
   Full step-by-step solution

   Step 1: Recall the z-score formula: z = (x - μ) / σ, where x is the data value, μ is the mean, and σ is the standard deviation. Step 2: Identify the given values: x = 22 minutes (Sophia's time), μ = 28 minutes (mean time), σ = 4 minutes (standard deviation). Step 3: Substitute into the formula: z = (22 - 28) / 4. Step 4: Calculate the numerator: 22 - 28 = -6. Step 5: Divide by the standard deviation: -6 / 4 = -1.5. Step 6: The z-score is -1.5, which means Sophia's time is 1.5 standard deviations below the mean. Since a lower time is better in running, this indicates her performance is significantly faster than the average runner.

4. A normal distribution has μ = 92 and σ = 9. Tane scores x = 119. Find the z-score. Answer: 3 Solution: Recall the z-score formula: z = (x - μ) / σ Substitute the given values: z = (119 - 92) / 9 Calculate the numerator: 119 - 92 = 27 Divide by the standard deviation: 27 / 9 = 3 The z-score is 3, meaning Tane's score is 3 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 = (119 - 92) / 9 Step 3: Calculate the numerator: 119 - 92 = 27 Step 4: Divide by the standard deviation: 27 / 9 = 3 Step 5: The z-score is 3, meaning Tane's score is 3 standard deviations above the mean. The answer is 3.

5. 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.

6. 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.

7. μ = 82, σ = 6, x = 97, 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, meaning the data point is 2.5 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 = (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, meaning the data point is 2.5 standard deviations above the mean. The answer is 2.5.