Questions: Homework Section 7.3 Question 2 of 10 (1 point) Question Attempt: 1 of Unlimited A sample of size 86 will be drawn from a population with mean 90 and standard deviation 24. Use the TI-83 Plus/TI-84 Plus calculator. Part 1 of 2 (a) Find the probability that x̄ will be less than 89. Round the answer to at least four decimal places. The probability that x̄ will be less than 89 is 0.1308.

Solution Steps

To find the probability that the sample mean \( \bar{x} \) is less than 89, we first calculate the Z-score for the upper bound:

\[ Z_{end} = \frac{X - \mu}{\sigma / \sqrt{n}} = \frac{89 - 90}{24 / \sqrt{86}} \approx -0.3864 \]

Next, we use the Z-score to find the probability. The probability that \( \bar{x} \) is less than 89 is given by:

\[ P(\bar{x} < 89) = \Phi(Z_{end}) - \Phi(-\infty) = \Phi(-0.3864) - 0 \]

Using the standard normal distribution table or calculator, we find:

\[ P(\bar{x} < 89) \approx 0.3496 \]

Final Answer