Dr. Nunez wants to estimate the average lung capacity of adults in her state. She plans on taking a sample of $n$ adults to construct a $99 \%$ confidence interval for the mean lung capacity. She wants the margin of error to be no more than $100 \mathrm{~mL}$. She's willing to use $\sigma=300 \mathrm{~mL}$ as an estimate for the population standard deviation.

Which of these is the smallest approximate sample size required to obtain the desired margin of error?

Choose 1 answer:
(A) 35 adults
(B) 45 adults
(C) 50 adults
(D) 55 adults
(E) 60 adults

Question

Dr. Nunez wants to estimate the average lung capacity of adults in her state. She plans on taking a sample of $n$ adults to construct a $99 \%$ confidence interval for the mean lung capacity. She wants the margin of error to be no more than $100 \mathrm{~mL}$. She's willing to use $\sigma=300 \mathrm{~mL}$ as an estimate for the population standard deviation.

Which of these is the smallest approximate sample size required to obtain the desired margin of error?

Choose 1 answer:
(A) 35 adults
(B) 45 adults
(C) 50 adults
(D) 55 adults
(E) 60 adults

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ To find the smallest sample size required, we use the formula for the sample size for estimating a population mean with a given margin of error (M), population standard deviation (σ), and confidence level (c). ‖ ‖ step 2 ⋮ The formula for the sample size (n) is given by $n = 2 \left(\frac{\sigma \cdot ext{erf}^{-1}(c)}{M} ight)^2$. Here, $ ext{erf}^{-1}(c)$ is the inverse error function corresponding to the confidence level. ‖ ‖ step 3 ⋮ Plugging in the values: $M = 100$, $\sigma = 300$, and $c = 99\%$, we calculate the sample size. ‖ ‖ step 4 ⋮ The calculation gives us a sample size of approximately 59.71, which we round up to the nearest whole number since we cannot have a fraction of a sample. ‖ ∻Answer∻ ⚹ The smallest approximate sample size required is 60 adults (Option E). ⚹ ∻Key Concept∻ ⚹Sample Size Calculation for Population Mean Estimation⚹ ∻Explanation∻ ⚹The sample size is calculated using the margin of error, population standard deviation, and confidence level to ensure that the estimate of the population mean is within the desired precision. The sample size must be rounded up to the nearest whole number as we cannot have a fraction of a sample.⚹◊What is the formula to calculate the sample size required to achieve a certain margin of error in a confidence interval for the mean?⍭ Generate me a similar question◊

Sia