A company wants to market a new chocolate bar. Its research department has come up with two versions, one with almonds and one with peanuts. The company intends to market the version with peanuts unless the majority of potential customers prefers almonds. To decide this, a taste test with a sample of 3600 potential customers was performed with the following results: 1999 favor the version with almonds, while 1601 favor the one with peanuts.
Assume all samples are simple random samples and α (alpha) is taken to be 0.05.
To decide which version of the chocolate bar to market, we need to conduct a hypothesis test.
Let's state our null and alternative hypotheses:
- Null hypothesis (H0): The majority of potential customers prefers the version with peanuts.
- Alternative hypothesis (Ha): The majority of potential customers prefers the version with almonds.
We will be using a one-sample proportion test to determine if the proportion favoring almonds is significantly different from the proportion favoring peanuts.
Given that we have a sample size of 3600 and that 1999 favor the version with almonds, we can calculate the sample proportion of customers favoring almonds (p-hat):
p-hat = (1999/3600) = 0.5553
Next, we need to calculate the standard error (SE) of the sample proportion:
SE = sqrt((p-hat*(1-p-hat)) / n)
SE = sqrt((0.5553*(1-0.5553)) / 3600)
SE = 0.0087
Now, we will conduct a one-sample proportion z-test to test the null hypothesis:
z = (p-hat - p) / SE
where p is the proportion specified in the null hypothesis. In this case, p = 0.5 (assuming equal preference for both versions).
z = (0.5553 - 0.5) / 0.0087
z = 6.32
At a significance level of α = 0.05, the critical z-value for a two-tailed test would be ±1.96. Since the calculated z-value (6.32) is much higher than the critical value, we reject the null hypothesis.
Therefore, we can conclude that the majority of potential customers prefer the version with almonds, and the company should market that version of the chocolate bar.