Probability and Statistical Inference

A scientist grows a species of plant in standard soil and the same species of plant in a soil that has been treated with nutrients. The scientist is trying to determine if the nutrient-treated soil will cause the plant to grow taller.

There are 5 plants in the standard soil. The heights of these plants, in cm, after one week are: 1.4, 1.8, 1.5, 2.1, 1.3

There are 5 plants in the nutrient-treated soil. The heights of these plants, in cm, after one week are: 1.5, 2.0, 1.7, 2.3, 1.6

1. Find the mean height of the plants in standard soil. Find the mean height of the plants in nutrient-treated soil. Then find the difference of the means. What does this difference mean in context?
2. The scientist randomly assigns the 10 plants to two different groups, and finds the difference of the means of the two groups. She runs this experiment several times and ends up with the data shown in the dot plot below.

Do you think there is evidence to support the conclusion that the nutrient-treated soil caused the plant to grow taller? Explain your reasoning. 

To see what might be happening when we regroup data, consider an experiment that takes 12 people and divides them into 2 groups at random. The control group contains 6 people and the treatment group contains 6 people. To explore what’s possible, assume the control group results in the data: 1, 3, 4, 6, 8, and 10. The treatment group results in the data: 2, 5, 7, 9, 11, and 12.

1. Find the difference in means for the original groups by subtracting the control group mean from the treatment group mean. 
2. With a smaller data set like this, we can actually consider all of the different arrangements of the data. There are 924 distinct ways to separate the 12 values into 2 groups of 6. The frequency table shows all the possible differences in means and how often they occur. Notice that a difference in means of 4.33 occurs 7 times and a difference of -4.33 also occurs 7 times. The dot plot shows the same information. 
   
   What proportion of possible groupings have a difference at least as great as the difference in means for the control group? Explain or show your reasoning. 
   
   
   

3. The proportion you calculate represents the probability that the original difference in means could be due to the groupings themselves. Based on the proportion you calculated for this situation, which description is most accurate? Explain your reasoning. 
   
   Description 1: Because the proportion is so low, it is unlikely that the difference in means is due to the randomized groupings. This means that the difference in means is most likely caused by the treatment. 
   
   Description 2: Because the proportion is not that low, it is still rather possible that the original difference in means is due to the random groupings. This means that there is not enough evidence to determine that the difference in means is likely caused by the treatment. 

A scientist divides 30 strawberry plants into two groups at random. One group of 15 plants will represent the control group and is grown in standard greenhouse conditions. The second group of 15 plants will represent the treatment group and will grow under the same conditions except they are grown in a different type of soil. After 6 weeks, the total weight (in grams) of the strawberries are measured for each plant. The scientist then performs a randomized experiment to compare the groups. 

The data are summarized by these statistics and histogram. 

• The mean for the control group is 238.67 grams
• The mean for the group with different soil is 347.47 grams

Is there evidence that the difference in means of the original groupings is due to the different soil, or is it likely that the difference is due to the way the plants were grouped? Explain your reasoning.