Question

An experiment on memory was performed, in which 16 subjects were randomly assigned to one of two groups, called "Sentences" or "Intentional". Each subject was given a list of 50 words. Subjects in the "Sentences" group were told to form multiple sentences, each using at least two words from the list, and to keep forming sentences until all the words were used at least once. Subjects in the "Intentional" group were told to spend five minutes memorizing as many of the 50 words as possible. Subjects from both groups were then asked to write down as many words from their lists as they could recall. The data are in the table below.

Number of words recalled

"Sentences" group	27	29	30	31	32	31	36	30
"Intentional" group	34	28	35	35	32	31	33	31

Enter this data into JMP in "long form" (e.g. each column should be a variable and each row should be an observation). We are interested in determining if there is a significant difference in the average number of words recalled for subjects in the "sentences" group vs. subjects in the "intentional" group, using α = 0.05. Use JMP to answer the questions below, and round all answers to three decimal places.

a. The appropriate null/alternative hypothesis pair for this study is:
(you have two attempts at this question)

1. H₀: μ_d = 0 ; H_A: μ_d ≠ 0

2. H₀: μ_d = 0 ; H_A: μ_d < 0

3. H₀: μ_sentences - μ_intentional = 0 ; H_A: μ_sentences - μ_intentional > 0

4. H₀: μ_sentences - μ_intentional = 0 ; H_A: μ_sentences - μ_intentional ≠ 0

5. Ho: μ_sentences - μ_intentional = 0 ; H_A: μ_sentences - μ_intentional < 0

6. H₀: μ_d = 0 ; H_A: μ_d > 0

b. Enter the values for the following statistics:
x_sentences = ______
s_sentences = ______
x_intentional = ______
s_intentional = ________
(x_sentences - x_intentional) = ________
standard error of (x_sentences - x_intentional) = _______ (Please explain how do you get the value for this)
test statistic: t = _______
p-value = _______

c. Report the 95% confidence interval JMP gives for μ_sentences - μ_intentional
Lower bound =  
Upper bound =  

d. From these results, our statistical conclusion should be:
(You have two attempts at this question.)

1. The means for "sentences" and "intentional" differ significantly, because the p-value is less than α and zero is inside the confidence interval

2. The means for "sentences" and "intentional" differ significantly, because the p-value is less than α and zero is outside the confidence interval    

3. The means for "sentences" and "intentional" differ significantly, because the p-value is less than α and -1.625 is inside the confidence interval

4. The means for "sentences" and "intentional" differ significantly, because the p-value is less than α and -1.625 is outside the confidence interval

5. The means for "sentences" and "intentional" do not differ significantly, because the p-value is greater than α and zero is inside the confidence interval

6. The means for "sentences" and "intentional" do not differ significantly, because the p-value is greater than α and zero is outside the confidence interval

7. The means for "sentences" and "intentional" do not differ significantly, because the p-value is greater than α and -1.625 is inside the confidence interval

8. The means for "sentences" and "intentional" do not differ significantly, because the p-value is greater than α and -1.625 is outside the confidence interval

Answer 1

Solution:-

a)

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis: u₁ = u ₂
Alternative hypothesis: u₁ u ₂

Note that these hypotheses constitute a two-tailed test. The null hypothesis will be rejected if the difference between sample means is too big or if it is too small.

Formulate an analysis plan. For this analysis, the significance level is 0.05. Using sample data, we will conduct a two-sample t-test of the null hypothesis.

Analyze sample data. Using sample data, we compute the standard error (SE), degrees of freedom (DF), and the t statistic test statistic (t).

SE = sqrt[(s₁²/n₁) + (s₂²/n₂)]
SE = 1.249
DF = 14
t = [ (x₁ - x₂) - d ] / SE

t = -1.301

where s₁ is the standard deviation of sample 1, s₂ is the standard deviation of sample 2, n₁ is the size of sample 1, n₂ is the size of sample 2, x₁ is the mean of sample 1, x₂ is the mean of sample 2, d is the hypothesized difference between the population means, and SE is the standard error.

Since we have a two-tailed test, the P-value is the probability that a t statistic having 14 degrees of freedom is more extreme than -1.301; that is, less than -1.301 or greater than 1.301.

Thus, the P-value = 0.214.

Interpret results. Since the P-value (0.214) is greater than the significance level (0.05), we have to accept the null hypothesis.

c) 95% confidence interval JMP gives for μ_sentences - μ_intentional is C.I = (- 4.304, 1.054).

C.I = (30.75 - 32.375) + 2.145*1.2491

C.I = - 1.625 + 2.67932

C.I = (- 4.304, 1.054)

d) 5. The means for "sentences" and "intentional" do not differ significantly, because the p-value is greater than α and zero is inside the confidence interval.