Question

Please explain and show all work:

A double-blind study was performed on consumers comparing a new formulation of a diet soft drink with the original formulation. Of 100 subjects, 57 preferred the new formulation, 38 preferred to original formulation, and 5 were undecided or could not tell the difference. Use the sign test to decide whether consumers prefer the new formulation at the 5% level.

Answer 1

The provided number of positive signs is and the number of negative signs is

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: Median of Differences = 0

Ha: Median of Differences > 0

The test statistic is

Observe that the sample size is large enough to use normal approximation, so a z-statistic will be used.

(2) Rejection Region

Based on the information provided, the significance level is α=0.05, and the critical value for a right-tailed test is zc​=1.64.

The rejection region for this right-tailed test is R={z:z>1.64}

(3) Test Statistics

The z-statistic is computed as follows:

(4) Decision about the null hypothesis

Since it is observed that , it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is p=0.9676, and since p=0.9676≥0.05, it is concluded that the null hypothesis is rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is rejected. Therefore, there is not enough evidence to claim that the population median of differences is greater than 0, at the 0.05 significance level. Hence there is not sufficient evidence to claim that consumers prefer the new formulation at the 5% level.

Graphically

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