Understanding Statistical Concepts: Sample Size, Causation, Confidence Intervals, and Z-Scores

1. The t-test is never valid if the sample size is small, say less than 30. (TRUE OR FALSE)

It can be valid if the sample is approximately normal.

2. The results of an observational study indicate that people who use vitamin supplements get fewer colds than people who don't. However, we can't conclude that vitamin supplements prevent colds because this type of study does not allow us to infer causation. (TRUE OR FALSE)

Only experimental designs can infer causation because it allows us to create treatment groups that are very similar.

3. A survey was administered to a random sample of college students. Both males and females were surveyed, and one question asked was "How much are you willing to spend on a stereo system (in dollars)?" An analysis was carried out to see if the average price students were willing to spend differs across gender. The 95% confidence interval for the true difference in means is given by:

100.86 ≤ μ_male - μ_female ≤ 386.62

Identify whether the following statements about this interval are correct by circling whether the statement is true or false.

a. We are 95% certain that, on average, college students are willing to spend between $100.86 and $386.62 on a stereo system. (TRUE OR FALSE)

This CI is for a difference in means, not one single mean.

b. The study provides statistical evidence that males are willing to spend more on a stereo system than females. (TRUE OR FALSE)

All the values in the CI are above 0.

c. The study provides evidence that 95% of all college students spend between $100.86 and $386.62 on a stereo system. (TRUE OR FALSE)

False for the same reason A is false: the CI is only for a difference in means.

d. We have evidence that 95% of males are willing to spend more on a stereo system than females. (TRUE OR FALSE)

This is not how we interpret/think about confidence intervals.

4. It is possible for the standard deviation to be less than zero. (TRUE OR FALSE)

The standard deviation takes the sum of squares; squared values cannot be negative, accordingly neither can their sum.

5. An observation that is close to the mean has a Z-score that is close to zero. (TRUE OR FALSE)

6. An observation that has a Z-score of 1 can be classified as a potential outlier. (TRUE OR FALSE)

A z-score must be roughly more extreme than +/- 2.

7. A study of malignant breast tumors was conducted at the University of Wisconsin-Madison. A sample of malignant tumor cells was examined under an electron microscope, and researchers measured the cell radius of each. They then obtained a 95% confidence interval for the true mean radius of malignant cells, which is given by 17.03 ≤ μ ≤ 17.90. Identify whether the following statements about this interval are correct by circling whether the statement is true or false.

a. We are 95% certain that the true average radius for malignant breast tumor cells is between 17.03 and 17.90. (TRUE OR FALSE)

b. We are 95% certain that the true average radius for all (including benign) breast tumor cells is between 17.03 and 17.90. (TRUE OR FALSE)

I can't make inferences about benign tumors using a sample of malignant tumors.

c. We are 95% certain that the average radius for malignant breast tumor cells in our sample is between 17.03 and 17.90. (TRUE OR FALSE)

The sample mean will always be in the middle of the CI.