An organization advocating for tax simplification has proposed the implementation of an alternative flat tax system to replace the existing Federal income tax. Featuring a very simple two-line tax form – How much money did you make? Send it In an attempt to identify the partisan nature of support for their proposal, the tax reformers have conducted a simple survey. They collected random samples of n1 = 120 Republican voters and n2 = 80 Democrat voters, polled the sampled respondents and noted for each group the number of voters who favor the flat tax proposal. The results of the survey are summarized in the table below. Political Affiliation Favor (X) Total (n) Proportion (X/n) Republican 90 120 p-hat1 = 90/120 = 0.75 Democrat 50 80 p-hat2 = 50/80 = 0.625 Total 140 200 p-hat = 140/200 = 0.700
An organization advocating for tax simplification has proposed the implementation of an alternative flat tax system to replace the existing Federal income tax.
Featuring a very simple two-line tax form –
- How much money did you make?
- Send it
In an attempt to identify the partisan nature of support for their proposal, the tax reformers have conducted a simple survey. They collected random samples of
n1 = 120 Republican voters and n2 = 80 Democrat voters, polled the sampled respondents and noted for each group the number of voters who favor the flat tax proposal. The results of the survey are summarized in the table below.
|
Political Affiliation |
Favor (X) |
Total (n) |
Proportion (X/n) |
|
Republican |
90 |
120 |
p-hat1 = 90/120 = 0.75 |
|
Democrat |
50 |
80 |
p-hat2 = 50/80 = 0.625 |
|
Total |
140 |
200 |
p-hat = 140/200 = 0.700 |
Required Parts
d. Our tax reformers now turn to a comparative analysis of Republican versus Democratic support for the proposed flat tax. Estimate the difference [Republican - Democrat] between the proportions of individuals who favor the flat tax proposal and develop a 95% confidence interval for the estimated difference.
e. Is there sufficient evidence, based upon the survey data, to conclude that the difference in proportions, [Republican - Democrat], that favor the proposed flat tax is significant (significantly different from zero?) Conduct your test at the α = 0.05 level of significance and report the p-value for your test. Be sure to identify the hypotheses to be tested and state your conclusion in managerial
a. Initial concerns are for assessing the level of support among Democrats in particular, so for now we focus on the Democrat line (the second line) of the survey summary table. Develop a 95% confidence
b. Our flat tax heroes, concerned with the precision (or possible lack thereof) of the estimated proportion of Democrats in favor, are considering how a follow up study might estimate more precisely the proportion of Democrats in favor of the flat tax. How large a sample of Democrats should be selected to estimate the proportion in favor to within plus-or-minus 0.05 with 95% confidence? Note that the already existing study summarized above provides a source of planning information for the contemplated future
c. The bottom-line issue, with respect to Democratic support for the flat tax, is whether or not Democrats, as a group, favor the flat tax. Do the survey data provide sufficient evidence to conclude that the proportion of Democrats favoring the flat tax exceeds 5? Conduct your test at the α = 0.05 level of significance and report the p-value for your test
*Illustration of the normal distribution curve, highlighting the area under the standard normal curve from the left towards \( Z_0 \).*
This table provides standard normal distribution cumulative probabilities, indicated as \( \Pr(Z < z_0) \). It lists the cumulative probabilities for the lower half of the standard normal distribution.
#### Explanation of the Table
- The table is organized with \( z_0 \) values as the row headers in the leftmost column, ranging from -3.5 to 0.0 in increments of 0.1.
- Across the top of the table, decimals from 0.00 to 0.09 represent additional increments, allowing precise lookup of probabilities for specific \( z_0 \) values.
- For each combination of row and column, the table presents the cumulative probability up to that \( z_0 \) value.
#### Table Entries
- **-3.5**:
- \( 0.00023 \) for \( Z_0 = -3.50 \)
- \( 0.00024 \) for \( Z_0 = -3.51 \)
- \( 0.00024 \) for \( Z_0 = -3.52 \)
- And so on...
- **0.0**:
- \( 0.5000 \) for \( Z_0 = 0.00 \)
- \( 0.4960 \) for \( Z_0 = 0.01 \)
- \( 0.4920 \) for \( Z_0 = 0.02 \)
- And so on...
This layout allows users to accurately determine the probability that a standard normal variable is less than a given \( z_0 \) value, a critical component in statistics for hypothesis testing and confidence interval estimation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffbfb787d-9ffb-47dc-83ae-238d4be0c5cf%2F0c2a07c2-90b6-4e69-85f9-07bd594e7c46%2Fxhq4nt_processed.jpeg&w=3840&q=75)
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