High School

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A distributor claims that only 4% of his apples are bruised. A buyer for a grocery store chain suspects that the true proportion *p* is greater than 4%. She selects a random sample of 30 apples to test the null hypothesis *H₀: p = 0.04* against the alternative hypothesis *Hₐ: p > 0.04*. Which of the following statements about conditions for performing a one-sample z test for the population proportion is correct?

(a) The test should not be performed because the Random condition has not been met.
(b) The test should not be performed because the Large Counts condition has not been met.
(c) We can't determine if the conditions have been met until we have the sample proportion *p*.
(d) All conditions for performing the test have been met.

Answer :

To decide if the conditions for performing a one-sample z test for the population proportion are met, we need to evaluate a few key factors:

  1. Random Condition: The sample should be randomly selected to ensure that it is representative of the population. The problem states that a random sample of 30 apples is selected, so the Random condition has been met.

  2. Large Counts Condition: This condition requires that both [tex]np[/tex] and [tex]n(1-p)[/tex] are greater than 10, where [tex]n[/tex] is the sample size and [tex]p[/tex] is the population proportion under the null hypothesis. In this case:

    • [tex]np = 30 \times 0.04 = 1.2[/tex]
    • [tex]n(1-p) = 30 \times (1 - 0.04) = 28.8[/tex]

    We see that [tex]np = 1.2[/tex], which is less than 10. Therefore, the Large Counts condition is not met, as both values must be greater than 10 to proceed with the test.

Based on the above points, the correct statement is:

(b) The test should not be performed because the Large Counts condition has not been met.

This conclusion means that due to the small expected number of bruised apples, a one-sample z test is not appropriate in this scenario, and other methods or larger samples may be needed to draw conclusions about the population proportion.

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