Question

7. Elliot has finally figured out that his veggie tacos had WAY too much cilantro. He finds a new recipe and invites 500 loyal customers to come and try it. 394 customers indicate that they like the new veggie taco better than the old recipe. Elliot wants to know if 75% of the population will prefer the new recipe and YOU are the savvy business statistician he's found to figure it out! May the odds be ever in your... flavor. Set up the null and the alternative hypotheses. b. Determine the test statistic. Determine the p-value. d. At 95% confidence, test to determine if more than 80% of the population will prefer the new recipe. a. c.

This is all the information we were provided with

Answer 1

Solution:-

a)

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

Null hypothesis: P = 0.75
Alternative hypothesis: P $\neq$ 0.75

Note that these hypotheses constitute a two-tailed test.

Formulate an analysis plan. For this analysis, the significance level is 0.05. The test method, shown in the next section, is a one-sample z-test.

Analyze sample data. Using sample data, we calculate the standard deviation (S.D) and compute the z-score test statistic (z).

$S.D=\sqrt{\frac{P(1-P)}{n}}$

S.D = 0.019365

b)

z = (p - P) /S.D

z = 1.96

where P is the hypothesized value of population proportion in the null hypothesis, p is the sample proportion, and n is the sample size.

Since we have a two-tailed test, the P-value is the probability that the z-score is less than -1.962 or greater than 1.962.

P-value = P(z < - 1.962) + P(z > 1.962)

Use z-calculator to find the p-values.

c)

P-value = 0.0249 + 0.0249

Thus, the P-value = 0.0498

Interpret results. Since the P-value (0.0498) is less than the significance level (0.05), we have to reject the null hypothesis.

From the above test we have sufficient evidence in the favor of the claim that 75% of the population will prefer the new recipe.

d)

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

Null hypothesis: P = 0.80
Alternative hypothesis: P > 0.80

Note that these hypotheses constitute a one-tailed test.

Formulate an analysis plan. For this analysis, the significance level is 0.05. The test method, shown in the next section, is a one-sample z-test.

Analyze sample data. Using sample data, we calculate the standard deviation (S.D) and compute the z-score test statistic (z).

$S.D=\sqrt{\frac{P(1-P)}{n}}$

S.D = 0.017889
z = (p - P) / S.D

z = - 0.6708

where P is the hypothesized value of population proportion in the null hypothesis, p is the sample proportion, and n is the sample size.

Since we have a one-tailed test, the P-value is the probability that the z-score is greater than - 0.6708

Thus, the P-value = 0.7488

Interpret results. Since the P-value (0.7488) is greater than the significance level (0.05), we failed to reject the null hypothesis.

From the above test we do not have sufficient evidence in the favor of the claim that more than 80% of the population will prefer the new recipe.