Final answer:
To find the probability that the sample mean is between 85 and 92 using a TI-83 Plus or TI-84 Plus calculator, calculate the z-scores for 85 and 92 and then use the normal distribution function to obtain the probabilistic range between the two z-scores.
Step-by-step explanation:
When working with the Central Limit Theorem (CLT), we need to understand how it affects the distribution of sample means. For a population with mean (μ) 90 and standard deviation (σ) 15, when samples of size n = 25 are drawn, according to the CLT, the distribution of the sample means will also be normal with a mean equal to the population mean (μ = 90) and a standard deviation equal to the population standard deviation divided by the square root of the sample size (σ/√n = 15/√25 = 15/5 = 3).
Probability of Sample Mean Between 85 and 92
To find the probability that the sample mean is between 85 and 92, we'd use the standard normal distribution (z-distribution) since the sample mean has been standardized. Here's how to calculate it on a TI-83 Plus/TI-84 Plus calculator:
- Calculate the z-scores for 85 and 92 using the formula z = (x - μ) / (σ/√n).
- For x = 85, z = (85 - 90) / 3 = -5/3 ≈ -1.67.
- For x = 92, z = (92 - 90) / 3 = 2/3 ≈ 0.67.
- Use the calculator's distribution function to find the probability P(-1.67 < Z < 0.67).
- On the calculator, access the distribution menu, usually by pressing 2nd and then VARS (for DISTR). Select '2:normalcdf' to enter the normal CDF function.
- Enter the z-scores as the lower and upper limits, respectively, with ∞ as the default values for the bounds. In this case, normalcdf(-1.67, 0.67).
- The result given by the calculator is the probability that the sample mean is between 85 and 92.