Question

For each of the following, identify whether the question is:

a) A 1-sample t-test problem, a.k.a. a one quantitative variable problem. While from a certain way of looking at things a matched pairs design is a form of one variable design, for this assignment, do NOT pick this option if you think it is a matched pairs.

b) A 2-sample t-test problem, a.k.a. a one categorical and one quantitative variable problem. Note that here we are only considering problems with two sets of individuals and two means to compare to each other.

c) A matched pairs design.

1. We want to know if brine shrimp are larger when they are grown in water with higher salinity. We have a random sample of 21 shrimp grown at medium salinity and a random sample of 16 shrimp grown at high salinity, with length in mm recorded for each animal.

2. We are curious about whether men or women are more affected by osteoporosis. A random sample of 20 male-female married couples was taken where all people were over age 50. Each person submitted to a bone density scan (g/cm2 ).

3. Sorghum is one of the most important cereal crops in the world. Silicon is thought to improve sorghum’s resistance to disease and drought by hardening the walls of root cells. Researchers grew 45 seedlings in a silicon-rich nutrient solution and 55 seedlings in a silicon-absent nutrient solution for five days and measured the root growth in mm for all seedlings. The prediction was that silicon-fed seedlings would have longer roots.

4. Alzheimer’s disease is degenerative brain disease which results in patients gradually losing their cognitive functions (among other negative impacts). It was established that people in the general population on average complete a puzzle in 9 minutes. A sample of suspected Alzheimer’s patients took on average 14 minutes to complete the puzzle with a standard deviation of 5 minutes. The researchers are attempting to find out if the sample of suspected Alzheimer’s patients varied statistically from 9 minutes.

5. Ophthalmologists are interested in figuring out whether intra-ocular pressure (IOP, or a measure of pressure inside peoples’ eyeballs, measured in mmHg) is dependent on central cornea thickness (CCT, measured in mm). In 10 people, CCT in both eyes was measured and each eye was designated either the higher CCT eye or the lower CCT eye. Then, IOP was measured for each eye and they compared the IOP measurements between the higher CCT eyes and the lower CCT eyes.

6. Reproductive choices has an entire suite of effects on the rest of animals’ lives. For instance, a scientist hypothesized that lactating bighorn sheep with lambs would have higher nutrition requirements than sheep without lambs and would thus need to feed more. She observed the % time spent feeding by ewes (female sheep) during a season they were feeding lambs and also recorded the same data for the same ewes in a subsequent year when they did not have a lamb.

7. Animals use different means to communicate with each other and to determine whether other individuals are members of the same species. Researchers were curious to determine if slugs could identify members of their own species by interacting with slime left behind by other slugs. They lined a plastic box with tree leaves and mucus of slug species A on the left side and mucus of slug species B on the right side. A series of 23 test slugs of species B were let loose in the box, and their movements were filmed for one hour. Data were analyzed to calculate the % time spent on the side with the conspecific mucus and the % time spent on the side with the other species’ mucus. It was assumed that if these two measurements were consistently unequal, the slugs could detect species from the mucus.

8. Physicians investigated the brain damage caused by mild head injuries, otherwise known as a bump on the head. They took 30 patients with mild closed head injuries and 30 individuals without injuries, and tested the peak velocity these patients could move their arm in response to a stimulus. They were curious if the group with head injuries had slower response times than the control group.

9. The same group of physicians extracted from the literature the population-level rate of mild head injuries. They wanted to know if high school hockey players experienced a higher than average rate of such injuries. They found epidemiological summary data which indicated mild head injury rates in hockey players in all 50 states.

10. Ecologists are often interested in domino effects that nutrients have in food chains. For instance, nutrients added to wetlands can impact growth of reeds, which can affect the insects which feed on the reeds. Scientists attempted to establish baseline data on nitrogen levels in reeds in two marshes; in each marsh they collected 6 plant samples, dried them, and analyzed each sample for nitrogen content (percentage of dry weight).

Answer 1

ONE SAMPLE T TEST:

The one sample t-test is a statistical procedure used to determine whether a sample mean is statistically different from a known or hypothesized population mean. It is also known as Single Sample t test. The variable used in this test is known as test variable where it is compared against a "test value", which is a known or hypothesized value of the mean in the population.

TWO INDEPENDENT T TEST:

The independent t-test, also called the two sample t-test, independent-samples t-test or student's t-test, is an inferential statistical test that determines whether there is a statistically significant difference between the means in two unrelated groups. Unrelated groups, also called unpaired groups or independent groups, are groups in which the observations or subject in each group are different.

In order to run an independent t-test, you need the following:

• One independent, categorical variable that has two levels/groups.
• One continuous dependent variable.

MATCHED PAIR DESIGN:

A matched pairs design is a special case of a randomized block design. It can be used when the experiment has only two treatment conditions; and subjects can be grouped into pairs, based on some blocking variable. Then, within each pair, subjects are randomly assigned to different treatments.

The paired sample t-test, sometimes called the dependent sample t-test, is a statistical procedure used to determine whether the mean difference between two sets of observations is zero. In a paired sample t-test, each subject or entity is measured twice, resulting in pairs of observations.

1) We want to know if brine shrimp are larger when they are grown in water with higher salinity. We have a random sample of 21 shrimp grown at medium salinity and a random sample of 16 shrimp grown at high salinity, with length in mm recorded for each animal.​

​In this question, we need to test whether the brine shrimp are larger when they are grown in water with higher salinity compared to medium salinity. Thus two different random samples of 21 shrimp grown at medium salinity ​and 16 shrimp grown at high salinity​ are used. Here we have one independent, categorical variable that has two levels/groups (medium & high salinity) and one continuous dependent variable Growth (length in mm). Thus this is a TWO SAMPLE T TEST PROBLEM.

​2) We are curious about whether men or women are more affected by osteoporosis. A random sample of 20 male-female married couples was taken where all people were over age 50. Each person submitted to a bone density scan (g/cm2 ).

In this question, we need to test whether the men or women are more affected by osteoporosis. Thus a random samples of 20 male-female married couples was taken. Here we have one independent, categorical variable that has two levels/groups (male & female) and one continuous dependent variable bone density(g/cm2). Thus this is a TWO SAMPLE T TEST PROBLEM.

3) Sorghum is one of the most important cereal crops in the world. Silicon is thought to improve sorghum’s resistance to disease and drought by hardening the walls of root cells. Researchers grew 45 seedlings in a silicon-rich nutrient solution and 55 seedlings in a silicon-absent nutrient solution for five days and measured the root growth in mm for all seedlings. The prediction was that silicon-fed seedlings would have longer roots.

In this question, we need to test whether the silicon-fed seedlings would have longer roots compared to silicon absent seedlings. Thus two different random samples of 245 seedlings in a silicon-rich nutrient solution ​and 55 seedlings in a silicon-absent nutrient solution​ are used. Here we have one independent, categorical variable that has two levels/groups (silicon rich & silicon absent) and one continuous dependent variable Growth of seedlings (in mm). Thus this is a TWO SAMPLE T TEST PROBLEM.

4) Alzheimer’s disease is degenerative brain disease which results in patients gradually losing their cognitive functions (among other negative impacts). It was established that people in the general population on average complete a puzzle in 9 minutes. A sample of suspected Alzheimer’s patients took on average 14 minutes to complete the puzzle with a standard deviation of 5 minutes. The researchers are attempting to find out if the sample of suspected Alzheimer’s patients varied statistically from 9 minutes.

The one sample t-test is a statistical procedure used to determine whether a sample mean is statistically different from a known or hypothesized population mean.​ Here one single sample of suspected Alzheimer’s patients are tested whether their average puzzle completion time is 9 minutes which is a hypothized value. Thus this is ONE SAMPLE T TEST.

5) Ophthalmologists are interested in figuring out whether intra-ocular pressure (IOP, or a measure of pressure inside peoples’ eyeballs, measured in mmHg) is dependent on central cornea thickness (CCT, measured in mm). In 10 people, CCT in both eyes was measured and each eye was designated either the higher CCT eye or the lower CCT eye. Then, IOP was measured for each eye and they compared the IOP measurements between the higher CCT eyes and the lower CCT eyes.

In this question, a single sample of 10 people were involved and CCT in their both eyes were measured and each eye was designated (paired) as either the higher CCT eye or the lower CCT eye. Thus 10 subjects are grouped into 2 pairs (higher CCT,lower CCT). Now IOP measurements were compared between the 2 pairs higher CCT eyes and the lower CCT eyes. Thus this is a MATCHED PAIR DESIGN.

6) Reproductive choices has an entire suite of effects on the rest of animals’ lives. For instance, a scientist hypothesized that lactating bighorn sheep with lambs would have higher nutrition requirements than sheep without lambs and would thus need to feed more. She observed the % time spent feeding by ewes (female sheep) during a season they were feeding lambs and also recorded the same data for the same ewes in a subsequent year when they did not have a lamb.

In this problem, we need to test whether lactating bighorn sheep with lambs would have higher nutrition requirements than sheep without lambs and would thus need to feed more. Thus % time spent feeding from the subject (female sheeps) were measured during a season they were feeding lambs​ and % time spent feeding is measured for the same set of subjects in a subsequent year when they did not have a lamb.​ Thus the subjects (female sheeps) are grouped into 2 pairs (season with lamb & season without lamb) and the % time spent feeding ​were compared between both pairs. Thus this is a MATCHED PAIR DESIGN.

7) Animals use different means to communicate with each other and to determine whether other individuals are members of the same species. Researchers were curious to determine if slugs could identify members of their own species by interacting with slime left behind by other slugs. They lined a plastic box with tree leaves and mucus of slug species A on the left side and mucus of slug species B on the right side. A series of 23 test slugs of species B were let loose in the box, and their movements were filmed for one hour. Data were analyzed to calculate the % time spent on the side with the conspecific mucus and the % time spent on the side with the other species’ mucus. It was assumed that if these two measurements were consistently unequal, the slugs could detect species from the mucus.

In this question, we need to test whether % time spent on the side with the conspecific mucus of slug species A and the % time spent on the side with the slug species B mucus Thus two different random samples of two species are used. Here we have one independent, categorical variable that has two levels/groups (Species A & Species B) and one continuous dependent variable (percentage time spent). Thus this is a TWO SAMPLE T TEST PROBLEM.

8) Physicians investigated the brain damage caused by mild head injuries, otherwise known as a bump on the head. They took 30 patients with mild closed head injuries and 30 individuals without injuries, and tested the peak velocity these patients could move their arm in response to a stimulus. They were curious if the group with head injuries had slower response times than the control group.

In this question, we need to test if the group with head injuries had slower response times than the control group. Thus two different random samples of 30 patients with mild closed head injuries ​and30 individuals without injuries are used. Here we have one independent, categorical variable that has two levels/groups (with head injury & without head injury) and one continuous dependent variable (response time). Thus this is a TWO SAMPLE T TEST PROBLEM.

9) The same group of physicians extracted from the literature the population-level rate of mild head injuries. They wanted to know if high school hockey players experienced a higher than average rate of such injuries. They found epidemiological summary data which indicated mild head injury rates in hockey players in all 50 states.

The one sample t-test is a statistical procedure used to determine whether a sample mean is statistically different from a known or hypothesized population mean.​ Here one single sample of hockey players in all 50 states are tested whether their mild head injury rates higher than average population-rate of such injuries. Thus this is ONE SAMPLE T TEST.

10) Ecologists are often interested in domino effects that nutrients have in food chains. For instance, nutrients added to wetlands can impact growth of reeds, which can affect the insects which feed on the reeds. Scientists attempted to establish baseline data on nitrogen levels in reeds in two marshes; in each marsh they collected 6 plant samples, dried them, and analyzed each sample for nitrogen content (percentage of dry weight).

In this question, we need to test on nitrogen levels in reeds in two marshes; Thus two different random samples of 2 marshes are used. Here we have one independent, categorical variable that has two levels/groups (2 different marshes) and one continuous dependent variable nitrogen levels (percentage of dry weight). Thus this is a TWO SAMPLE T TEST PROBLEM.