College

A farmer knows that a grocery store will reject a shipment of his vegetables if more than 4% of the vegetables contain blemishes. He inspects a large truckload of tomatoes to determine if the proportion with blemishes (p) exceeds 0.04. He selects an SRS of 150 tomatoes from the more than 2,000 tomatoes in the truck. Suppose that 8 tomatoes sampled are found to have blemishes. Which of the assumptions for inference about a proportion is violated, if any?

a. Large Counts: [tex]np > 10[/tex]
b. Large Counts: [tex]n(1 - p) > 10[/tex]
c. The sample is a random sample of the entire population.
d. 10% condition: the sample size is less than 10% of the population.
e. There do not appear to be any violations.

Answer :

Answer:

Assumption A is violated

Step-by-step explanation:

x = number of blemishes = 8

n = sample size = 150

proportion = x/n = 8/150 = 0.0533

1. large counts: np> 10

= 150 * 0.0533 >10

= 7.995 > 10

this assumpton is obviously violated. 7.995 is not greater than 10

2. Large Counts: n(1 - p) > 10

150(1-0.0533)>10

150-7.995 > 10

142.005> 10

there is no violation. The assumption is satisfied

3. This assumption is satisfied. this is because the tomatoes were selected using simple random sampling.

4. 10% of the population = 0.1 * 2000 = 200

the sample size = 150

150 < 200

this assumption is satisfied

Final answer:

The 'Large Counts' assumption for np > 10 is violated when a farmer inspects a random sample of 150 tomatoes and finds 8 with blemishes, since the value of np is only 6, not greater than 10.

Explanation:

The question from the student involves determining if any inference about a proportion assumptions are violated when a farmer checks for blemishes on his tomatoes. The assumptions checked are based on a set of conditions necessary for valid statistical testing of proportions.

  1. Large Counts: The np and n(1-p) should both be greater than 10 for the normal approximation to be reliable.
  2. A sample should be a random sample from the population.
  3. The 10% condition states the sample size must be less than 10% of the population in order to avoid dependence within the sampling process.

Given that the farmer has a simple random sample (SRS) of 150 tomatoes, and only 8 tomatoes have blemishes, we calculate np and n(1-p) as follows:

  • np = 150 * 0.04 = 6
  • n(1-p) = 150 * (1 - 0.04) = 150 * 0.96 = 144

A violation is found in the 'Large Counts' condition for np, as np should be greater than 10 but is only 6 in this case. Thus, answer choice (a) is the correct assumption that is violated.

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