Question

What is the conjecture that the economist is trying to find support for?

Suppose that he takes a sample of 265. Which would give more support to his conjecture – finding that 204 students received financial aid or that 215 students received financial aid? Explain your choice.

Assuming the newspaper’s claim is correct, find the probability of observing 204 or more students on financial aid and the probability of finding 215 or more students on financial aid. How do the two probabilities relate to the answer you gave to the previous question?

0.840 .810 .780 .750 .720 .690 .660 .63 Sample Proportion

0.900 .870 .840 .810 .780 .750 .72 Sample Proportion

According to an article on webmd.com, 28.6% of Kentucky residents smoked in 2000. After significant advertising campaigns by the American Cancer Society, a researcher would like to know if the proportion of smokers has decreased. A random sample of 672 Kentucky residents is taken, and each is asked whether or not they smoke.

What is the conjecture that we would like to find evidence for?

Completely describe the sampling distribution for the sample proportion of Kentucky residents that smoke when samples of size 672 are taken.

Suppose that a sample of 672 residents is taken, and 27.9% smoke. Assuming the webmd value of 28.6% is still accurate, what is the probability of observing a smoking percentage of 27.9% or less in this sample?

Based on the probability, what (if anything) can be inferred about the true percentage of KY residents

Step 1

**Note:**

Hi! Thank you for posting the question. Since you have posted several questions, we have solved all parts of the first question (survey by professor regarding financial aid) for you. If you need any particular question to be answered, please mention it and post the question again.

Step 2

**Central limit theorem for proportions:**

Suppose a population has true proportion of “success”, or proportion of cases favourable to the category of interest, *π*. A sample of size *n* is taken from this population. Then, for a sufficiently large sample size, the probability distribution for the sample proportion of success (*p*) is normal, with *μ* = *π* and *σ* = √[* π* (1 – *π*) / n].

The central limit theorem can be applied only if the following assumptions and conditions hold:

*Randomization:*The samples must be collected randomly from the same population, so that each individual in the sample has the same distribution as the population.*10% condition:*The sample size must be at most 10% of the population size.*Large enough sample:*The sample size*n*should be large enough, so that*nπ*≥ 10 and*n*(1 –*π*) ≥ 10.

The 2005 newspaper article claims that 75% students got financial aid, that is, *π* = 0.75.

The number of students considered by the professor is *n* = 265. It can be assumed that the selected students are representative of all the students considered, that is, come from the same population.

Now, each year, a large number of college students seek financial aid. So, it is logical to assume that *n* = 265 is not greater than 10% of all students.

Here, *nπ* = (265) ∙ (0.75) = 198.75 >10, *n *(1 – *π*) = (265) ∙ (1 – 0.75) = (265) ∙ (0.25) = 66.25 > 10.

Thus, all the conditions of the Central Limit Theorem are satisfied. So, the Central Limit Theorem can be applied to describe the sampling distribution for the model of the sample proportion of students seeking financial aid. The distribution is normal with mean *μ* = 0.75 and standard deviation *σ* = √[0.75) ∙ (0.25) / 265] ≈ 0.0266.

Step 3

**The suitable distribution:**

It is assumed that the newspaper claim made in 2005, that the percentage of students requiring financial aid is 75%, still holds true. As described in the previous part, the mean is *μ* = 0.75.

The first figure shows a distribution that is approximately bell-shaped, like the normal ...

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