A local school board believes there is a difference in the proportion of households with school-aged children that would support starting the school year a week earlier, and the proportion of households without school-aged children that would support starting the school year a week earlier.

They survey a random sample of 40 households with school-aged children about whether they would support starting the school year a week earlier, and 38 households respond yes. They survey a random sample of 45 households that do not have school-aged children, and 35 respond yes.

Let \( p_s \) be the true proportion of households with school-aged children that would support starting the school year a week early, and \( p_w \) be the true proportion of households without school-aged children that would support starting the school year a week earlier.

Which of the following is a correct statement about the conditions for this test?

A. The random condition is not met.
B. The 10% condition is not met.
C. The Large Counts Condition is not met.
D. All conditions for inference are met.

Firstly, the random condition is satisfied as the households are selected randomly from the population, ensuring that the sample is representative of the population.

Secondly, the 10% condition is met since the sample sizes (40 households with school-aged children and 45 households without) are each less than 10% of their respective populations, ensuring that the sampling does not significantly affect the population.

Finally, the Large Counts Condition is satisfied as both the expected counts (38 and 35) are greater than 10, ensuring that the sampling distribution of sample proportions is approximately normal, allowing for valid inference using normal distribution assumptions.

These conditions ensure that the sample accurately represents the population, the sampling process doesn't unduly influence results, and the sampling distribution approximates a normal distribution, allowing for reliable statistical inference regarding the difference in proportions of households supporting starting the school year earlier between those with and without school-aged children.

Therefore, all conditions for inference are met in this test.

JoeEthan JoeEthan · Answer 2 · 2024-01-28T14:07:50-05:00

Final answer:

Assuming that the surveyed households were randomly selected and the sizes of the groups surveyed are much less than 10% of the entire household population, the random condition and the 10% condition are met. The Large Counts Condition is also met, with both samples having counts greater than 5 for np and n(1-p). Therefore, all conditions for inference are met.

Explanation:

In the scenario where a local school board is comparing the proportion of households with school-aged children that support starting the school year a week earlier to the proportion of households without school-aged children who support the same, there are a few conditions that must be met for statistical inference testing. The question is asking which of the conditions is not met:

Random Condition: The samples must be a random selection from the population. If the households were randomly selected, then this condition is met.
10% Condition: The samples should be less than 10% of the population to use certain approximation methods. Typically, households in a survey would be well below this threshold.
Large Counts Condition: This condition requires that both np and n(1-p) are greater than 5 for each sample. We can check this by using the sample proportions (38/40 for households with children and 35/45 for households without).

After calculating, we find that both 38/40 and 35/45 are greater than 5, indicating that the Large Counts Condition is met. Therefore, assuming the samples were randomly selected, and given that the sample sizes are typically much smaller than 10% of the entire population of such households, we can conclude that all conditions for inference are met for this test.