Question

In Exercises 55–60, find the indicated probabilities and interpret the results.

The mean annual salary for physical therapists in the United States is about \$87,000. A random sample of 50 physical therapists is selected. What is the probability that the mean annual salary of the sample is (a) less than \$84,000? Assume sigma = \$10,500.

Accepted Answer

Step 1: Identify the key parameters of the problem. The population mean (μ) is $$87,000, the population standard deviation (σ) is 10$$,500, the sample size (n) is 50, and we are tasked with finding the probability that the sample mean is less than $$84,000. Step 2: Recall the formula for the standard error of the mean (SEM), which is used to measure the variability of the sample mean. The formula is: SEM = σn. Substitute σ = 10,500 and n = 50 into the formula to calculate the SEM. Step 3: Standardize the sample mean using the z-score formula for a sampling distribution: z = X̄ - μSEM. Here, X̄ = 84$$,000, μ = $$87,000, and SEM is the value calculated in Step 2. Substitute these values into the formula to compute the z-score. Step 4: Use the z-score obtained in Step 3 to find the corresponding probability. Refer to the standard normal distribution table (z-table) or use statistical software to determine the cumulative probability associated with the z-score. Step 5: Interpret the result. The cumulative probability represents the likelihood that the sample mean annual salary of 50 physical therapists is less than 84$$,000. Express this probability as a percentage and provide context for the result.

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Concetti chiave

Central Limit Theorem

Standard Error

Z-Score