Question

Suppose that two populations have the same mean. A researcher draws a sample of size 35 from each population and calculates the difference in sample means. He then repeats this process 99 more times, resulting in 100 calculated differences in sample means. The researcher finds the standard error of the difference in sample means to be 1.78. Which of the following statements is true regarding the distribution of the differences in sample means? The center of the distribution will be approximately 0, with about 68 percent of the differences in means between -3.56 and 3.56

Answer 1

Explanation:

Hello!

You have the difference between two sample means. Remember the sample means are variables with certain distribution, let's say, in this case, both sample means have a normal distribution. X[bar] = sample mean

X₁[bar]~N(μ₁;σ₁²/n₁)

X₂[bar]~N(μ₂;σ₂²/n₂)

Then the difference between these two variables results in a third variable that will also have a normal distribution:

X₁[bar]-X₂[bar]~N(μ₁-μ₂;σ₁²/n₁+σ₂²/n₂)

These are some of the properties of the normal distribution

Centered in μ

Symmetrical

Bell-shaped

[μ - σ; μ + σ]= 68% of the distribution

[μ - 2σ; μ+ 2σ]= 95% of the distribution

[μ - 3σ; μ+ 3σ]= 99.7% of the distribution

Check attachment.

Taking these properties into account, if you where to draw the results of the 100 trials, where μ₁-μ₂=0 would be its center and the standard deviation of the difference is 1.78.

68% of the information will be between (μ₁-μ₂) ± [(σ₁/√n₁)+(σ₂/√n₂)], this is 0 ± 1.78

I hope it helps!