Question

The Environmental Protection Agency (EPA) has standards and regulations that says that the lead level in soil cannot exceed the limit of 400 parts per million (ppm) in public play areas designed for children. Mike is an inspector, and he takes 50 randomly selected soil samples from a site where they are considering building a playground.
These data show a sample mean of x = 390.25 ppm and standard deviation of Sx = 30.5 ppm. Answer the following:

1. Determine the critical t value and along with the confidence interval for 95% level of

confidence. 2. What does this interval suggest?

Answer 1

Answer:

See below

Explanation:

Check One-Sample T-Interval Conditions

Random Sample? √

Sample Size ≥30? √

Independent? √

Population Standard Deviation Unknown? √

One-Sample T-Interval Information

• Formula --> $CI=\bar{x}\pm t^*(\frac{S_x}{\sqrt{n}})$
• Sample Mean --> $\bar{x}=390.25$
• Critical Value --> $t^*=2.0096$ (given $df=n-1=50-1=49$ degrees of freedom at a 95% confidence level)
• Sample Size --> $n=50$
• Sample Standard Deviation --> $S_x=30.5$

Problem 1

The critical t-value, as mentioned previously, would be $t^*=2.0096$, making the 95% confidence interval equal to $CI=\bar{x}\pm t^*(\frac{S_x}{\sqrt{n}})=390.25\pm2.0096(\frac{30.5}{\sqrt{50}})\approx\{381.5819,398.9181\}$

This interval suggests that we are 95% confident that the true mean levels of lead in soil are between 381.5819 and 398.9181 parts per million (ppm), which satisfies the EPA's regulated maximum of 400 ppm.