Consider the folllowing competing hypotheses accompanying sample data drawn inde

Question

Consider the folllowing competing hypotheses accompanying sample data drawn independently:

Ho: 1 - 2 3

H1: 1 - 2 > 3

Sample Mean

(FORMATTING: Negative values should be indicated by a minus sign, no spaces! Positive/Negative values should use the following format: "=-0.00". Round intermediate calculations to 6 decimal places and final answer to 2 decimal places).

a. Calculate the critical value using a 90% confidence level:

b. Calculate the value of the test statistic:

c. Calculate the p-value:

d. Do you have enough evidence to reject Ho at alpha level?

(type only "yes" or "no").

Description Population 2 Population 1

Sample Mean

73 79 Sample Standard Deviation 11 1.68 Sample Size 16 16

Explanation / Answer

Solution:-

The solution to this problem takes four steps: (1) state the hypotheses, (2) formulate an analysis plan, (3) analyze sample data, and (4) interpret results. We work through those steps below:

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

Null hypothesis: 1 - 2< 3
Alternative hypothesis: 1 - 2 > 3

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

Formulate an analysis plan. For this analysis, the significance level is 0.10. Using sample data, we will conduct a two-sample t-test of the null hypothesis.

Critical value. Using significance level of 0.10 (90% confidence) is -1.28 (left tailed) or 1.28(right tailed)

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 = sqrt[(1.682/16) + (112/16)] = 2.782

DF = (s12/n1 + s22/n2)2 / { [ (s12 / n1)2 / (n1 - 1) ] + [ (s22 / n2)2 / (n2 - 1) ] }
DF = (1.682/16 + 112/16)2 / { [ (1.682 / 16)2 / (15) ] + [ (112 / 16)2 / (15) ] }
DF = 59.891 / (0.002074464 + 3.81276041667) = 15.70

t = [ (x1 - x2) - d ] / SE = [(79 - 73) - 3] / 2.782 = 1.08

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 population means, and SE is the standard error.

Here is the logic of the analysis: Given the alternative hypothesis (1 - 2 > 3), we want to know whether the observed difference in sample means is large enough (i.e., sufficiently more than 3) to cause us to reject the null hypothesis.

We use the t Distribution Calculator to find P(t < 1.08)

The P-Value is 0.148227.
The result is not significant at p < 0.10

Interpret results. Since the P-value (0.15) is greater than the significance level (0.10), we cannot reject the null hypothesis.

Conclusion. Fail to reject null hypothesis. We do not have sufficient evidence to reject the null hypothesis at the given significance level.

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