Takehome Portion Examples


The following examples illustrate the use of the MA145 technology link with the type of problems that appear on the takehome portion of the exam.

The problems fall into two major categories:

Most of the problems on the exam deal with hypothesis testing.

The hypothesis testing problems fall into two subcategories:

Example 1

A researcher counts gypsy moth (Lymantria dispar) egg masses in 76 quarter-acre plots. An much larger survey from the previous season indicated an average of 52.1 egg masses per quarter-acre with a standard deviation of 12.4. If the average number of egg masses per quarter-acre in the sample is 46.1, does the data support the claim that the level of infestation (as measured by the average number of egg masses per quarter acre) is lower this year?

Solution: Note the following when reading the problem:

In general, the null hypothesis can be almost anything, but by convention it is usually associated with the situation where nothing has changed: the treatment had no effect, there was no change in the level of a measured variable, the mean this year is the same as last year.

In this case, the likely choice for a null hypothesis would be a statement contradicting the claim that the infestation is lower:

Null Hypothesis H_0: The mean number of egg masses per quarter acre 52.1, the same as last year.

Alternative Hypothesis H_1: The mean number of egg masses per quarter acre is less than 52.1.

The wording for a comparison of two means and a test of a hypothesis about a mean are similar, but the fact that only one sample size is given suggests there is only one sample mean, which would be the case in testing a hypothesis about a mean. The value of the mean under the null hypothesis does not come from a sample, but is purely hypothetical.

So we would conclude that:

A check of the MA145 technology page for hypothesis testing, sigma known lists the required inputs as the mean under H_0, the sample mean, the (known) population standard deviation sigma, the sample size, and the alpha level (which is chosen by the person analyzing the data).

From the wording of the claim, we would perform a one-sided test, "left tailed" because we want to reject the null hypothesis for sufficiently small values of the sample mean, and we do not care about sample means higher than the mean under the null hypothesis.

Example 2

Ocean color is known to correlate strongly with phytoplankton levels, green indicating higher and blue lower levels of phytoplankton. Ocean color in the Gulf of Alaska is measured on a grid of 120 randomly selected coordinate points using satelite imagery from 2005. The same analysis is performed on an lower resolution image taken 3 years earlier, using 80 randomly selected coordinate points. In the 2005 data, the wavelength of the peak absorption color is 492.1 nanometers with a standard deviation of 43.2. The 2002 data has peak absoption at 511 nanometers with a standard deviation of 32.4. Given that the wavelength of green light is 510 nm and that of blue light is 475 nm, does this data indicate a significant shift in ocean color towards blue (and therefore, a reduction in phytoplankton)?

Solution: Note the following when reading the problem:

In general, the null hypothesis can be almost anything, but by convention it is usually associated with the situation where nothing has changed: the treatment had no effect, there was no change in the level of a measured variable, the mean this year is the same as last year.

In this case, the likely choice for a null hypothesis would be a statement contradicting the claim that there is a shift in ocean color towards blue:

Null Hypothesis H_0: There is no significant difference between the peak absorption wavelengths in 2002 and 2005 (i.e., the two populations, 2002 and 2005, have the same mean)

Alternative Hypothesis H_1: The peak absorption wavelength is lower in 2005 (the population means are different, and the 2005 mean is lower)

The wording for a comparison of two means and a test of a hypothesis about a mean are similar, but the fact that we are given two sample means, two apparent sample standard deviations, and two sample sizes suggests that this problem can be handled as inference about two means, with sigma unknown.

So we would conclude that:

A check of the MA145 technology page for inference about two means, sigma unknown lists the required inputs as the two sample means, the two sample standard deviations, and the two sample sizes (and also the alpha level, which is not based on data but chosen by the data analyst).

Example 3

Mosquito traps are placed near 43 small ponds and a count of Culex species in the traps is obtained during a baseline period. At the end of the baseline period a spraying program is conducted. One week after the spraying, the traps are cleaned and a second collection period is initiated. Based on an estimate of the size of each pond, the raw counts are converted to a density of Culex mosquitoes per square foot of pond. The difference between the before and after densities is found to have a sample mean of 9.1 and a sample standard deviation of 12.0. Test whether or not the data indicates that the spraying was effective in reducing the density of Culex species mosquitoes.

Solution: Note the following when reading the problem:

In this case, the likely choice for a null hypothesis would be a statement that the spraying has no effect:

Null Hypothesis H_0: The mean population densities of mosquitoes is the same before and after spraying (i.e., spraying is not effective).

Alternative Hypothesis H_1: The mean population density is lower after spraying (spraying is effective).

The wording for a comparison of two means and a test of a hypothesis about a mean are similar, but the fact that we are given two sample means, two apparent sample standard deviations, and two sample sizes suggests that this problem can be handled as inference about two means, with sigma unknown.

So we would conclude that:

A check of the MA145 technology page for inference about two means with paired samples lists the required inputs as the mean difference between paired measures, the standard deviation of the difference, and the sample size (number of pairs). (plus, the alpha level).

Example 4

In a double-blind study of Major Depressive Disorder (MDD), 43 subjects are treated with seratonin-reuptake inhibitors (SRIs) while 32 are given a placebo. After 8 weeks of treatment, 16 subjects in the SRI group have experienced a remission of MDD, while 6 subjects in the placebo group have. Can we conclude that patients receiving the drug are more likely to remit than those receiving a placebo?

Solution: Note the following when reading the problem:

The data provided suggests that this is inference about two proprotions.

Null Hypothesis H_0: There is no difference between the proportion of subjects with MDD who remit when treated with SRIs and the proportion who remit when given a placebo.

Alternative Hypothesis H_1: A higher proportion of the MDD subjects treated with SRIs remit.

So we would conclude that:

A check of the MA145 technology page for inference about two proportions lists the required inputs as the two proportions, and the two sample sizes. (plus, the alpha level).

Example 5

A government health agency estimates the level of HIV infection in a certain area at 15.3%. Testing is performed on a random sample of 45 residents and 12 of the tests are positive. Does this data support the agency figure?

Solution: Note the following when reading the problem:

The data provided suggests that this is a test of a hypothesis about a proportion.

Null Hypothesis H_0: The HIV infection level is 15.3%.

Alternative Hypothesis H_1: The HIV infection level is different from 15.3%.

So we would conclude that:

A check of the MA145 technology page for tests of hypotheses on proportions lists the required inputs as the proportion when the null hypothesis is true, the sample proportion, and the sample size. (plus, the alpha level).