Question
1 2 3 4 5 6 7 8 9 TIME REMAINING 41:29 Dion is performing a hypothesis test in which the population mean is 92 and the standard deviation is 2 . His sample size is 7 with a mean of 93.5 . Which of the following correctly depicts the \( z \)-statistic for this data? 0.28 0.36 1.98 2.63
Ask by Donnelly Huang. in the United States
Feb 25,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The z-statistic is approximately 1.98.
Solution
To compute the z-statistic, use the formula:
z = (x̄ - μ) / (σ/√n)
where:
• x̄ = sample mean = 93.5
• μ = population mean = 92
• σ = population standard deviation = 2
• n = sample size = 7
Step 1: Compute the numerator (the difference between the sample mean and the population mean):
93.5 - 92 = 1.5
Step 2: Compute the denominator (the standard error):
σ/√n = 2 / √7
Step 3: Write the z-statistic:
z = 1.5 / (2/√7)
Simplify by multiplying numerator and denominator appropriately:
z = 1.5 * (√7 / 2)
Approximating √7 ≈ 2.6458:
z ≈ 1.5 * 2.6458 / 2
z ≈ 3.9687 / 2
z ≈ 1.98435
Rounded, z ≈ 1.98.
Thus, the correct answer is 1.98.
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Simplify this solution The Deep Dive
To calculate the \( z \)-statistic for Dion's data, we can use the formula: \[ z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}} \] Where: - \(\bar{x}\) is the sample mean (93.5) - \(\mu\) is the population mean (92) - \(\sigma\) is the standard deviation (2) - \(n\) is the sample size (7) First, calculate the standard error (SE): \[ SE = \frac{2}{\sqrt{7}} \approx 0.756 \] Then, plug the values into the \( z \)-statistic formula: \[ z = \frac{93.5 - 92}{0.756} \approx \frac{1.5}{0.756} \approx 1.98 \] So, the correct \( z \)-statistic for Dion's data is **1.98**.
