A random sample is selected from a normal population with a mean of __40 and a standard deviation of __10. After a treatment is administered to the individuals in the sample, the sample mean is found to be M _ 42.

a. How large a sample is necessary for this sample mean to be statistically significant? Assume a two tailed test with _ = .05
b. If the sample mean were M _ 41, what sample size is needed to be significant for a two-tailed test with _ = .05.

mean 100 standard deviation 20 population 1000

To determine the sample size necessary for the sample mean to be statistically significant, we need to calculate the required sample size based on the given information and hypothesis testing parameters.

a. To find the required sample size when the sample mean is M = 42, we need to use the formula for calculating the sample size for a two-tailed test. The formula is given as:

n = [(Zα/2 * σ) / E]^2

where:
n = required sample size
Zα/2 = critical value for a two-tailed test at the specified significance level (α/2 = 0.025 for α = 0.05)
σ = standard deviation of the population
E = acceptable margin of error (measured in terms of the standard deviation)

In this case, the given standard deviation (σ) is 10 and the acceptable margin of error (E) is unknown. To determine the margin of error, we need to consider the critical value and the difference between the sample mean (M) and the population mean (μ).

Given that our two-tailed test with α = 0.05, we can find the critical value using a Z-table or a statistical software. At α/2 = 0.025, the critical value (Zα/2) is approximately 1.96 for a standard normal distribution.

Now that we have the critical value and the population standard deviation, we can calculate the margin of error (E) using the formula:

E = Zα/2 * (σ / sqrt(n))

Substituting the given values, we have:

E = 1.96 * (10 / sqrt(n))

To find the required sample size (n), we need to rearrange the equation and solve for n:

n = (Zα/2 * σ / E)^2
n = (1.96 * 10 / E)^2

Since the acceptable margin of error (E) is not given, we cannot calculate the exact sample size without that information. You would need to specify the acceptable margin of error to calculate the required sample size.

b. To determine the required sample size when the sample mean is M = 41, we follow the same steps as in part (a) to calculate the margin of error (E) and the sample size (n).

Given that the margin of error (E) is unknown, we need to calculate it using the same formula as before:

E = Zα/2 * (σ / sqrt(n))

n = (Zα/2 * σ / E)^2

However, in this case, the difference between the sample mean (M) and the population mean (μ) is different. The difference is calculated as:

Difference = |M - μ| = |41 - 40| = 1

Now, we need to find a critical value (Zα/2) that corresponds to a two-tailed test with α = 0.05 and a difference of 1. We can use a Z-table or statistical software to find this critical value. Let's assume it is approximately 1.96.

Substituting the values into the formula, we have:

n = (1.96 * 10 / E)^2

Again, since the acceptable margin of error (E) is not given, we cannot calculate the exact sample size without that information. You would need to specify the acceptable margin of error to calculate the required sample size.