Question

Step 3: Assess the Evidence

We have summarized the sample data with a sample proportion. We now determine the strength of our evidence with a P-value. Remember, the P-value is the probability of observing a statistic that is at least as extreme as the one we gathered, assuming the null hypothesis is true.

Since the criteria for approximate normality are satisfied, we can use the normal distribution to determine the P-value.

A. Notice that the normal curve below is centered at the assumed population proportion, p=0.61. This is because the mean of the sampling distribution of sample proportions is the population proportion. The population proportion is assumed to be p=0.61 based on the null hypothesis. On the image below, plot the sample proportion computed (pihat = 0.66)

Graph:

Mean=0.61

Standard diviation = 0.02

B. The P-value is the probability of randomly observing a sample proportion at least as extreme (in this case, larger) as the one gathered in our sample, assuming that the null hypothesis is true. Shade the area under the normal curve above that corresponds to this probability.

C. To find the area shaded above we use the standard normal distribution. This requires that we compute the z-score of the sample proportion above. To compute the z-score, we need the mean and standard error of the sampling distribution of sample proportions.

Find the mean and standard error of the sampling distribution of sample proportions (use three decimal places for the standard error). We have an assumed value for the population proportion (p) null hypothesis.

Mean = u pi hat = p ___________

Standard error = sigma pi hat = square root of (p(1-p))/n = _______________________

D. We can use the mean and standard error to compute the Z-score of our pi hat value.

Z=(pi hat - u pi hat)/sigma pi hat) = ____________________

E. Use technology or tables to find the normal probability of observing a Z-score that is greater than or equal to the one computed. This is the P-value.

P-value = P (pi hat >0.66)=P(Z>_____________)=___________________

F. The null hypothesis assumes that there is no change in the proportion of adults in the U.S. (p=0.61) who believe that upper-income Americans pay too little in taxes. But, we observed a simple proportion (pi hat = 0.66) that was greater. Does the P-value indicate that the sample proportion we observed was likely or unlikely, given a population proportion of 0.61?

Step 4: State a conclusion:

Now that we have computed a P-value we can make a decision

A. We have stated that when a P-value is less than our 5% level of significance, the sample proportion is statistically significant. This leads us to believe that null hypothesis is unlikely to be true. Is the observed sample proportion (pi hat = 0.66) statistically significant? What decisions should we make about the null and alternative hypothesis?

Remember, in conclusion, we can support or fail to support the alternative hypothesis, but we never conclude that the null hypothesis is true. The only way to prove the null hypothesis is to sample the entire population!

B. Describe what this means with regard to the proportion of all adults in the U.S. in 2017 who believe that upper-income Americans pay too little in taxes.

...other values given in the exercise: Null hypothesis: p=0.61; Alternative hypothesis: p>0.61; pi hat = 0.66: np = 305; n(1-p) = 195 (This values were calculated in the previous steps: Step 1 - Determine the Hypotheses, Step 2 - Collect the Data)

Answer 1

A.

We know, population proportion .The sample proportion is plotted as follows.

B.

The area corresponding to P-value is as follows.

C.

Mean = p = 0.61

It is given that np = 305 and we have p = 0.61 and so n = 305/0.61 = 500

Standard error

D.

Our test statistic is given by

So, our Z score is given by

E.

   [Using R-code '1-pnorm(1.438707)']

F.

Level of significance

We reject our null hypothesis if

Here, we observe that .

So, we fail to reject null hypothesis.

Hence, we fail to support the alternative hypothesis.

Based on the given data we can conclude that, the proportion of all adults in the U.S. in 2017 who believe that upper-income American pay too little in taxes is not significantly greater than 0.61.