Question

You need to compute the 95% confidence interval for the population mean. How large a sample should you draw to ensure that the sample mean does not deviate from the population mean by more than 2.2?

Answer 1

To calculate the sample size needed for a 95% confidence interval so that the sample mean does not deviate from the population mean by more than 2.2, the standard deviation of the population must be known. Using the margin of error formula and the z-score for 95% confidence, the sample size can be derived.

Step-by-step explanation:

To determine how large a sample should be to ensure that the sample mean does not deviate from the population mean by more than 2.2 with 95% confidence, we need to use the formula for the margin of error in a confidence interval:

E = z * (σ/√n)

Where E is the maximum error of the estimate (in this case, 2.2), z is the z-score corresponding to the desired confidence level (for 95%, it's typically 1.96), σ is the standard deviation of the population, and n is the sample size.

Since the standard deviation of the population (σ) is not provided, it is assumed to be known or estimated from previous studies. Solving for n, we get:

n = (z * σ / E)^2

Without the standard deviation, it's impossible to calculate the exact sample size needed. Assuming σ is known, you would plug in the values for z and σ and solve for n to ensure the sample mean is within 2.2 units of the population mean with 95% confidence.

Sample size calculation is a critical aspect of designing a study, ensuring that the results are statistically significant and reliable for drawing conclusions about the population mean. The confidence interval is a range within which we expect the true population parameter to fall with a certain level of confidence, reflecting the precision of our estimate.

Answer 2

To calculate the required sample size, we need to use the formula:

n = (Z * σ / E)²

Where:

n is the required sample size.

Z is the Z score corresponding to the desired confidence level (95%).

σ is the standard deviation of the population (unknown).

E is the maximum allowed error (2.2).

Since the standard deviation (σ) of the population is unknown, we can estimate it from a previous study or pilot study. If there are no previous studies available, we can use a conservative estimate of 0.5, which assumes the largest possible variability. Keep in mind that a higher standard deviation estimation would yield a larger required sample size.

Assuming σ = 0.5 and a 95% confidence level, the Z score corresponding to a 95% confidence level is approximately 1.96.

Plugging the values into the formula, we have:

n = (1.96 * 0.5 / 2.2)²

n = (0.98 / 2.2)²

n = 0.445²

n ≈ 0.198

Rounding up to the nearest whole number:

n = 1

To ensure that the sample mean does not deviate from the population mean by more than 2.2, you should draw a sample size of at least 1. It is important to note that this is a theoretical minimum, and in practice, it is not possible to have a sample size of 1. In real-world scenarios, you would typically need a larger sample size to generate reliable results.

Step-by-step explanation: