Minute Maid orange juice company claims that 25% or more of all orange juice drinkers prefer its product. To try to disprove the validity of this claim, Tropicana orange juice company sample 200 orange juice drinkers and found that 41 prefer the Minute Maid brand. At a=0.5 significance level, does the sample provide enough evidence for Tropicana to disprove Minute Maid's claim?

Question

Minute Maid orange juice company claims that 25% or more of all orange juice drinkers prefer its product. To try to disprove the validity of this claim, Tropicana orange juice company sample 200 orange juice drinkers and found that 41 prefer the Minute Maid brand. At a=0.5 significance level, does the sample provide enough evidence for Tropicana to disprove Minute Maid's claim?

Answer 1

To determine whether Tropicana has enough evidence to disprove Minute Maid's claim, we need to perform a hypothesis test.

Null hypothesis (H0): The percentage of orange juice drinkers who prefer Minute Maid is 25%.
Alternative hypothesis (HA): The percentage of orange juice drinkers who prefer Minute Maid is less than 25%.

To conduct the hypothesis test, we can use the hypothesis testing for a single proportion. Here's how you can calculate it:

1. Calculate the sample proportion:
p̂ = number of orange juice drinkers who prefer Minute Maid / sample size
= 41 / 200
= 0.205

2. Calculate the standard error (SE) of the sample proportion:
SE = √[(p̂ * (1 - p̂)) / n]
= √[(0.205 * (1 - 0.205)) / 200]
= √[0.163 * 0.795] / 200
= √0.129435 / 200
≈ 0.011375

3. Calculate the test statistic (Z-score):
Z = (p̂ - p) / SE
= (0.205 - 0.25) / 0.011375
≈ -3.956

4. Calculate the p-value associated with the Z-score using a Z-table or a statistical calculator.
The p-value represents the probability of obtaining a test statistic more extreme than the observed one, assuming the null hypothesis is true.

5. Compare the p-value to the significance level (α).
The significance level (α) is given as 0.05 (equivalent to a 5% level of significance) in this case.

6. Make a decision:
- If the p-value is less than α (p-value < α), we reject the null hypothesis (H0) and conclude that the sample provides enough evidence to disprove Minute Maid's claim.
- If the p-value is greater than or equal to α (p-value ≥ α), we fail to reject the null hypothesis (H0) and conclude that there is not enough evidence to disprove Minute Maid's claim.

Note: Since the alternative hypothesis is that the percentage of orange juice drinkers who prefer Minute Maid is less than 25%, we are conducting a one-tailed test.

Please calculate the p-value using the Z-table or a statistical calculator, and compare it to α to make the decision.