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------------------------------------------------ A farmer knows that a grocery store will reject a shipment of his vegetables if more than [tex]$4\%$[/tex] of the vegetables contain blemishes. He inspects a large truckload of tomatoes to determine if the proportion with blemishes ([tex]p[/tex]) exceeds 0.04. He selects a simple random sample (SRS) of 150 tomatoes from the more than 2,700 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]n \cdot p \ > \ 10[/tex]

B. Large Counts: [tex]n(1-p) \ > \ 10[/tex]

C. The sample is a random sample of the entire population.

D. [tex]10\%[/tex] condition: the sample size is less than [tex]10\%[/tex] of the population.

E. There do not appear to be any violations.

To determine if any assumptions for inference about a proportion are violated in this scenario, we need to assess several conditions:

1. Calculate the Sample Proportion ([tex]$ \hat{p} $[/tex]):
- Total number of tomatoes sampled ([tex]$ n $[/tex]) = 150
- Number of blemished tomatoes in the sample ([tex]$ x $[/tex]) = 8
- Sample proportion ([tex]$ \hat{p} $[/tex]) is calculated as:
[tex]\[
\hat{p} = \frac{x}{n} = \frac{8}{150} = 0.0533
\][/tex]

2. Population Proportion ([tex]$ p $[/tex]):
- The population proportion given is [tex]$ p = 0.04 $[/tex].

3. Large Counts Condition:
- This condition is fulfilled if both [tex]$ np > 10 $[/tex] and [tex]$ n(1-p) > 10 $[/tex].

- Check if [tex]$ np > 10 $[/tex]:
[tex]\[
np = 150 \times 0.04 = 6
\][/tex]
Since 6 is not greater than 10, this condition is violated.

- Check if [tex]$ n(1-p) > 10 $[/tex]:
[tex]\[
n(1-p) = 150 \times (1 - 0.04) = 150 \times 0.96 = 144
\][/tex]
Since 144 is greater than 10, this part of the condition is satisfied.

4. Random Sample:
- The sample is described as an SRS (Simple Random Sample) of the truckload, suggesting that the sample is random. This condition is assumed to be satisfied.

5. 10% Condition:
- The sample size must be less than 10% of the population for this condition to be satisfied.
- Population size = 2700 tomatoes
- Calculate 10% of the population:
[tex]\[
0.10 \times 2700 = 270
\][/tex]
Since the sample size of 150 is less than 270, this condition is satisfied.

Based on these checks, the violation occurs with the large counts condition for [tex]$ np $[/tex] since it is not greater than 10. Therefore, this assumption is violated.