Question

1. For the following two datasets labeled y₁ and y₂ match one quantity in column A with one quantity in column B. The sample means and variances are labeled as y1, y2, S12$gCjW8nT3nm55gAAAABJRU5ErkJggg==$ and S22. The population means and variances from which they were drawn are labeled μ1,μ2, σ12,$AAAAABJRU5ErkJggg==$ and σ22. Assume that the two samples are independent random samples.

2. 
   H₀: μ1=μ2 against the alternative H_a: μ1≠μ2 $pJu8OerEcQAAAABJRU5ErkJggg==$using significance level α=.01. Using the data from problem 1 provide the following information to test the hypothesis

2. 1. Formula for the test statistic:
   2. Value of the test statistic:
   3. Reference distribution (include the degrees of freedom):
   4. Critical value:
   5. Conclusion:
3. Using the data from problem 1, provide the following information to construct and interpret a 95% confidence interval for the mean of y₁.
   1. Formula for the confidence interval:
4. 2. Values of the lower and upper endpoints of the confidence interval:
5. 3. Does the confidence interval provide information to answer the question of whether the population mean of y₁ is equal to zero? If so, what does the confidence interval tell you?
6. 4. If the sample size were larger, would the width of the confidence interval change, and if so, how would it change?
7. 5. Which of the following correctly describes a 95% confidence interval for a mean? Circle the correct answer.
      1. A range within which 95% of all possible sample means fall
      2. An interval constructed using a procedure such that 95% of intervals constructed this way will contain the population mean.
      3. A range within which 90% of all data values in the population fall
      4. All of the above
      5. None of the above

need answers for all 3 questions

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$\neq$ u ₂

Note that these hypotheses constitute a two-tailed test.

Formulate an analysis plan. For this analysis, the significance level is 0.01. 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 = 5.2111
a) t = [ (x₁ - x₂) - d ] / SE

b) t = - 0.832

c)

D.F = n₁ + n₂ - 2

D.F = 6 + 6 - 2

D.F = 10

d) t_critical = + 3.17

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 10 degrees of freedom is more extreme than -0.832; that is, less than -0.832 or greater than-0.832.

Thus, the P-value = 0.425.

Interpret results. Since the P-value (0.425) is greater than the significance level (0.01), we have to accept the null hypothesis.

e) From the above test we do not have sufficient evidence in the favor of the claim that there is difference in the two means.