Imagine that a researcher appropriately collects data from 83 people from a particular population for variable X. They obtain a mean of 40.3 from these 83 people. He also knows that the population mean for variable X is 49 (μ = 49) with a standard deviation of 7.3 (σ = 7.3). Using this information, compute the z statistic.

Please report the correct z statistic below with two decimal places (if relevant).

Question

Imagine that a researcher appropriately collects data from 83 people from a particular population for variable X.  They obtain a mean of 40.3 from these 83 people.  He also knows that the population mean for variable X is 49 (μ = 49) with a standard deviation of 7.3  (σ = 7.3). Using this information, compute the z statistic.

Please report the correct z statistic below with two decimal places (if relevant).

BlackTom AI · Accepted Answer

We start by identifying the formula for the z statistic in a one-sample z test when the population standard deviation is known: z = (x̄ - μ) / (σ / √n).

First, compute the standard error of the mean: SE = σ / √n = 7.3 / √83.
√83 is approximately 9.11, so SE ≈ 7.3 / 9.11 ≈ 0.801.

Next, compute the difference between the sample mean and the population mean: x̄ - μ = 40.3 - 49 = -8.7.

Now form the z statistic with these values: z = (-8.7) / 0.801 ≈ -10.86.

Rounding to two decimal places as requested, the z statistic is approximately -10.86.

Key checks:
- The sign is negative because the sample mean is below the population mean.
- The standard error is the standard deviation of the sampling distribution of the mean, which decreases with larger sample size, hence the fairly large magnitude of z given a relatively small SE.

Note: This calculation assumes that σ = 7.3 is the known population standard deviation and that a z-test is appropriate for this scenario.

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