Question

Sampling Distribution of the Sample Proportion According to a 2005 newspaper report about financial aid for college students, 75% of all full-time degree-seeking students receive some form of financial aid. Given the recent financial crisis, an economist conjectures that the percentage receiving some form of financial aid has increased. In order to test his conjecture, he plans to sample 265 full-time degree-seeking students to determine the proportion that receive financial aid. of the two distributions below, which do you believe corresponds to the sampling distribution of the proportion of students who have financial aid assuming the newspaper claim is still correct? Explain your choice. Prep Using the properties discussed in class, describe the sampling distribution of the proportion of full-time degree-seeking students who receive some form of financial aid. 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 tnbuti...pd distribut...pdf × I D sampling dis × sampling di Χ | D sampling di, Χ。sampl ngdi X | s/dinehyl/Downloads/sampling%20distribution%20of%20p%20(2).pdf Assuming the newspoper'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? 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? Ca distrib of D sampling dis samplingdi: × D sampling dix | D sampling d x D sampl sampling di x sampling de X san /dinehy1/Downloads/sampling%20distribution%20of%20p%20(2).pdf According to an article on webmd.com 286%of Kentucky residents smoked in 2000. After sign cant 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? can be inferred about the true percentage of KY residents that smoke? sampling distnibuti. pdf sampling distnibuti sampling distribut... pdt ch

Answer 1

The first one where the newspaper claimed proportion comes in the middle of the graph ie exactly to the center could be the sampling distribution of proportion because the distribution is symmetrical about the point 0.75.

Hence this can be assumed to be normally distributed with mean 0.75 and standard error will be calculated below

The economist believes that the second graph could be the distribution of sampling proportion as the there is a shift to the higher side in the overall proportion of degree students seeking grants.

The economist suspects that the proportion of students receiving grants has increased so 215 students (215/265=0.81) receiving financial aid gives more support to his conjecture than 204 students among 265 which around 0.77

Given that the news paper claim is correct it means p =0.75

We will next find the standard deviation of proportion to find the probabilities which is given by $\sqrt{p(1-p)/n}=\sqrt{0.75*0.25/256}=0.0265$

Assuming normal distribution for this sampling proportion, normal with mean .75( since newspapers claim is assumed to be correct) and standard deviation 0.0265

Probability of observing more than 204 students=Probability of observing a proportion more than 0.77

=P(p>0.77)

$=P(z>\frac{0.77-0.75}{0.0265})=P(z>0.7547)=0.2252$

Probability of observing more than 215 students=Probability of observing a proportion more than 0.81

=P(p>0.81)

$=P(z>\frac{0.81-0.75}{0.0265})=P(z>2.264)=0.0118$

If the newspapers claim is believed to be correct probability of observing more than 215 students (1.18%) is even less than 2% that is negligible small which makes the economist conjecture wrong.