Elementary Statistics – Review Examples

Read each scenario, and then decide on the proper statistical analysis. (Be sure to check any assumptions of the analysis.)

1. Ozone, a prevalent photochemical oxidant, has been linked to forest decline and severe crop loss. A random sample of 50 ozone concentrations was obtained on Mt. Mitchell in Yancey County, North Carolina. An environmentalist is interested in estimating the mean ozone concentration for this area.

2. An Appleton resident is interested in predicting the amount of energy consumed in a home, based on the size of the home. He takes a sample of 30 homes, and for each he measures the size (in square feet) and the energy consumed (in kilowatt-hours per month).

3. A fair coin is to be flipped 10 times. A student wonders what the probability is of getting at least 8 heads.

4. A group of 40 women in a large city were given instructions on self-defense. Prior to the course, they were tested to determine their self-confidence. After the course they were given the same self-confidence test. The self-defense teacher wonders if self-confidence scores are higher on average after taking the self-defense course.

5. A consumer research group sampled 100 hand-held video games, all of the same make and model. The group wants to estimate the mean life span of the video games. Based on their previous work, they know the standard deviation of the population of life spans is 35 hours.

6. A doctor looks at 50 two-child families (one boy, one girl). For each of the families she measures the heights of the sister and the brother. She is interested in the association between the two sets of heights.

7. A fair coin is to be flipped 500 times. A student wonders what the probability is of getting no more than 240 heads.

8. A researcher has 50 rats. He randomly divides them into two groups of 25: group 1 rats receive a drug and group 2 rats receive a placebo. Then he measures the number of errors the rats make in a run through a maze. He wonders if there is a difference between the two groups, with regards to the average number of maze errors made.

Elementary Statistics – Answers to Review Examples

1. We can estimate the mean ozone concentration with a confidence interval. Because we don’t know the population standard deviation, we must use the one-sample t confidence interval. The sample size is large (n = 50 > 40), so even if the sample distribution of ozone concentrations is severely skewed, we can still use the t confidence interval (we should graph the sample data to make sure there are no outliers, though). There are 49 degrees of freedom for the t distribution.

2. If we are interested in predicting one variable from another, we can use a regression line (if appropriate). Hence, we can create a scatterplot (y-variable: energy consumed; x-variable: size of house) and, if appropriate, fit a regression line to the data.

3. The number of heads in 10 flips of a fair coin follows a binomial distribution with n = 10 and p = 0.5. Hence, we can use Table C to find the probability of at least 8 heads (add up the probabilities of 8, 9, and 10 heads).

4. This is a matched pairs design and the population standard deviation is unknown, so we should use a paired t-test. The sample size is large (n = 40), so even if the data distribution of score differences is severely skewed, we can still use the paired t-test (we should graph the sample data to make sure there are no outliers, though). There are 39 degrees of freedom, and the alternative is one-sided.

5. We can estimate the mean life span with a confidence interval. Since the population standard deviation is known, we should use the one-sample z confidence interval. The sample size is large (n = 100), so the confidence interval will be approximately correct, even if the original population isn’t normal (since the central limit theorem tells us the distribution of the sample mean is approximately normal).

6. To determine the association between the variables, we can create a scatterplot and calculate the correlation (to measure the strength and direction of the linear relationship).

7. The number of heads in 500 flips of a fair coin is binomial with n = 500 and p = 0.5. We can’t use Table C to find probabilities (since n > 20, and Table C doesn’t go beyond n = 20), but we can use the normal approximation. Since np = 250 and n(1-p) = 250 (which are both at least 10), the normal approximation should be good. Then the number of heads in 500 flips follows an approximate normal distribution, and we can use this distribution to calculate the probability of no more than 240 heads.

8. This is a two-sample (not a paired) data problem. The population standard deviations are unknown, so we should use a two-sample t test. The combined sample size (50) is large, so even if the data distributions of maze errors are severely skewed, we can still use the t-test (we should graph the sample data to make sure there are no outliers, though). There are 24 degrees of freedom (smaller sample size minus one), and the alternative is two-sided.