The calculation of the probability and interpretation of the difference in mean and the relation to level of significance between and the treatment and control groups are as follows;
Question 1; Part A
The probability that the treatment group mean is greater than the control group mean by 8 points or more is 3.6%
The completed statement is as follows;
The significance level is set at 5%, and the probability of the result is 3.6%, which is lower than significance level. The result is statistically significant
Part B
The result is statistically significant, which implies that spraying the corn plant with the new type of fertilizer does increase the growth rate
Question 2; Part A
The probability of the treatment group mean is lower than the control group mean by 15 points or more is 7.8%
The completed statement is as follows;
The significance level is set at 5%, and the probability of the result is 7.8% which is higher than the significance level. The result is not statistically significant
Part B;
The result is not statistically significant which implies that wearing a watch does not make people manage their time better
The reason for the above solutions is as follows;
Question 1;
The known parameters;
The significance level of the test, α = 5%
The frequencies of the difference in mean between the treatment and control groups given in the table
The number of randomizations, ∑f = 1,000
The required parameter
The probability of the treatment group mean is greater than the control group mean by 8 points or more
The significance of the result
The comparison between the given level of significance and the probability
Method
We find the probability for the required difference in mean from the data given in the table
The probability that the treatment group mean is greater than the control group mean by 8 points or more is given by the following formula;
[tex]\mathbf{P(8 \ or \ more) = \dfrac{\sum f(8 \ or \ more)}{\sum f}}[/tex]
From the table, we have;
∑f(8 or more) = 26 + 8 + 2 = 36
∑f = 1 + 10 + 28 + 58 +...+ 114 + 57 + 28 + 8 + 2 = 1,000
Therefore;
[tex]P(8 \ or \ more) = \dfrac{36}{1,000} = 0.036 = 3.6\%[/tex]
The probability, P(8 or more) 3.6%
Given that the significant level, α = 5% is higher than the probability of 3.6%, the result is statistically significant
Part B
The true statement is that the result is statistically significant, which implies that spraying the corn plant with the new type of fertilizer does increase the growth rate
Question 2;
The known parameters are;
The significance level of the test, α = 5%
The frequencies of the difference in mean between the treatment and control groups given in the table
The number of randomizations, ∑f = 1,000
The required parameter
The probability of the treatment group mean is lower than the control group mean by 15 points or more
The significance of the result
The comparison between the given level of significance and the probability
Method
Using the method from Question 1, we have;
Part A
[tex]\mathbf{P(15 \ or \ more) = \dfrac{\sum f(15 \ or \ less)}{\sum f}}[/tex]
From the table, we have;
∑f(15 or more) = 50 + 18 + 10 = 78
∑f = 10 + 18 + 50 + 127 + 195 + 226 + 180 + 117 + 48 + 17 + 12 = 1,000
Therefore;
[tex]P(15 \ or \ more) = \dfrac{78}{1,000} = 0.078 = 7.8\%[/tex]
The probability, P(15 or more) = 7.8%
Given that the significant level, α = 5% is lower than the probability of 7.8%, the result is not statistically significant
Therefore, we have;
The significance level is set at 5%, and the probability of the result is 7.8% which is higher than the significance level. The result is not statistically significant
Part B
The true statement is that the result is not statistically significant which implies that wearing a watch does not make people manage their time better
Learn more about hypothesis testing here;
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