If ${\mu }_X=3.2$ and ${\sigma }_X=0.98$, find the probability of getting a sample mean above 3.5 in a sample of 60 people.

• A. 0.009
• B. 0.018
• C. 0.003
• D. 0.006

Answer: (A)

Verified step by step guidance

1. Identify the given values: The population mean (μ_X) is 3.2, the population standard deviation (σ_X) is 0.98, and the sample size (n) is 60. We are tasked with finding the probability of the sample mean being greater than 3.5.
2. Calculate the standard error of the mean (SE) using the formula: $\mathrm{SE}=\frac{\mathrm{\sigma \_X}}{\sqrt{n}}$. Substitute the values of σ_X = 0.98 and n = 60 into the formula.
3. Standardize the sample mean to a z-score using the formula: $z=\frac{\mathrm{X̄}-\mathrm{\mu \_X}}{\mathrm{SE}}$. Here, X̄ = 3.5, μ_X = 3.2, and SE is the value calculated in the previous step.
4. Use the z-score obtained to find the cumulative probability from the standard normal distribution table. This gives the probability of the sample mean being less than 3.5.
5. To find the probability of the sample mean being greater than 3.5, subtract the cumulative probability from 1. This final value represents the probability of the sample mean exceeding 3.5.