Final answer:
To find the number of observations needed to estimate a population mean at a 95% confidence interval with a given standard error, use the formula n = (σ / SEM)2, where σ is the standard deviation. The calculation results in n ≈ 111, and since we cannot have a fraction of an observation, we choose the next highest option, which is 200 observations.
Step-by-step explanation:
The question asks how many observations must be included in a sample to estimate a population mean with a sampling distribution error SE = 0.24, at 95% confidence interval, given that the population variance σ2 is approximately equal to 6.4. To solve this, we use the formula for the standard error of the mean (SEM), which is SEM = σ / √n, where σ is the standard deviation of the population and n is the sample size.
First, we find the standard deviation by taking the square root of the variance: σ = √6.4 = 2.53.
Next, rearranging the SEM formula to solve for n we get n = (σ / SEM)2. Plugging in the given values: n = (2.53 / 0.24)2 ≈ 111.02. Since we cannot have a fraction of an observation, we round up to the nearest whole number, which gives us n = 111. Thus, option b) 100 is not enough, and the next closest option is c) 200, which would suffice for our requirement.
Therefore, the correct answer is c) 200 observations.