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9. A candy company produces fruit candy pieces in four different flavors: lemon, orange, strawberry, and cherry. The company claims that each flavor is equally likely to appear in a package. A statistics student thinks that the true proportion \( p \) of cherry-flavored candies is less than 0.25. She takes a random sample of 25 candies to test the null hypothesis \( H_0: p = 0.25 \) against \( H_a: p < 0.25 \).

Which of the following statements about conditions for performing a one-sample z-test for the population proportion is correct?

A. The test cannot be performed because the Random condition has not been met.
B. The test cannot be performed because the Large Counts (Normal) condition has not been met.
C. The test cannot be performed because the Central Limit Theorem (Normal) condition has not been met.
D. All conditions for performing the test have been met.

Answer :

Final answer:

In this statistics problem, the random condition for performing the one-sample z test for population proportion is met because a random sample is taken. However, the large counts (normal) condition isn't met as we don't expect at least 10 successes and 10 failures in our sample. Therefore, the one-sample z test can't be performed. So, the correct option is B.

Explanation:

This problem deals with Statistics and hypothesis testing. To perform a one-sample z test for the population proportion, two key conditions must be met: the Random condition and the Large Counts or Normal condition.

(a) The Random condition is met as it's specified that a random sample of 25 candies is taken from the population.

(b) The Large Counts or Normal condition requires that we expect to find at least 10 instances of both success and failure in our sample.

Here, the expected number of successes (cherry flavored candies) and failures (other flavors) under the null hypothesis both exceed 10 (0.25*25=6.25 which is not greater than 10). Hence, this condition is not met.

Therefore, the correct answer is (b) The test cannot be performed because the Large Counts (Normal) condition has not been met.

Learn more about One-Sample Z Test here:

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