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. Which version should the company market?
Assume all samples are simple random samples and α (alpha) is taken to be 0.05.
To determine which version the company should market, we can conduct a hypothesis test using the data from the taste test.
Null Hypothesis (H0): The majority of potential customers prefer the version with peanuts.
Alternative Hypothesis (H1): The majority of potential customers prefer the version with almonds.
We will use a one-sample proportion test, comparing the proportion of customers who prefer almonds (1999 out of 3600) to the proportion who prefer peanuts (1601 out of 3600).
To perform the test, we can use a z-test for proportions. The test statistic is given by:
z = (p1 - p2) / sqrt(p*(1-p)*(1/n1 + 1/n2))
where p1 and p2 are the proportions, p is the pooled proportion, and n1 and n2 are the sample sizes.
First, we need to calculate the pooled proportion, p:
p = (x1 + x2) / (n1 + n2)
= (1999 + 1601) / (3600 + 3600)
= 0.5553
Next, we can calculate the test statistic, z:
z = (0.5553 - 0.4447) / sqrt(0.5553 * 0.4447 * (1/3600 + 1/3600))
= 5.6108
To determine whether to reject or fail to reject the null hypothesis, we compare the test statistic to the critical value at the significance level of α = 0.05.
The critical value for a two-tailed test at α = 0.05 is approximately ±1.96.
Since the test statistic, 5.6108, is greater than 1.96, we reject the null hypothesis. This means that the majority of potential customers prefer the version with almonds.
Therefore, the company should market the version with almonds.