Question

Last year, a soft drink manufacturer had 21% of the market. In an effor to increase their portion of the market, the manufacturer has introduced a new flavor in their soft drinks. A sample of 400 individuals participated in the taste test and 100 indicated that they like the taste. We are interested in determining if more than 21% of the population will like the new soft drink.

a. Set up the null and the alternative hypotheses.
b Determine the test statistic.
c. Determine the p-value.
d. At a 5% level of significance, test to determine if more than 21% of the population will like the new soft drink. Clearly state your decision and conclusions using the p-value approach.

Answer 1

To determine if more than 21% of the population will like the new soft drink, you need to set up null and alternative hypotheses, calculate the test statistic (z-score), determine the p-value by comparing the test statistic to the standard normal distribution table, and test at a 5% level of significance by comparing the p-value to the significance level (0.05).

Step-by-step explanation:

a. Set up the null and alternative hypotheses:

The null hypothesis, denoted as H0, is that more than 21% of the population will like the new soft drink. The alternative hypothesis, denoted as H1, is that 21% or less of the population will like the new soft drink.

b. Determine the test statistic:

The test statistic for this hypothesis test is the z-score. The formula to calculate the z-score is:

z = (p - P) / sqrt((P * (1 - P)) / n)

Where p is the sample proportion (100/400), P is the hypothesized proportion (0.21), and n is the sample size (400).

c. Determine the p-value:

The p-value is the probability of observing a test statistic as extreme as the one calculated, assuming the null hypothesis is true. To find the p-value, we would compare the test statistic (z-score) to the standard normal distribution table.

d. Test at a 5% level of significance:

To test the hypothesis at a 5% level of significance, we compare the p-value to the significance level (0.05). If the p-value is less than 0.05, we reject the null hypothesis in favor of the alternative hypothesis. Otherwise, we fail to reject the null hypothesis.

Based on the calculated test statistic and comparison to the standard normal distribution table, you can determine the p-value. Then, compare the p-value to the 0.05 significance level to make a decision and draw conclusions.