Inferential Statistics

Question

A vote is to be taken among the residents of a town and the surrounding county to determine whether a proposed chemical plant should be constructed. The construction site is within the town limits, and for this reason many voters in the count believe that the proposal will pass because of the large proportion of town voters who favor the construction.to determine if there is a significant difference in the proportions of town voters and county voters favoring the proposal, a poll is taken. If 120 of 200 town voters favor the proposal and 240 of 500 county residents favor it, would you agree that the proportion of town voters favoring the proposal is higher than the proportion of county voters? Use level of significance 0.05. Using the five-step procedure results 1. State the null and alternative hypothesis 2. State the Level of significance/ degree of freedom 3. Use the appropriate test statistics 4. State the critical or rejection region 5. Make your conclusion and recommendation

Answer 1

Hey Annika, this appears to be a two sample z test of proportions. To explain this fully would require more time and space than is available here but I will give you enough to answer the question and you can feel free to contact me for a more thorough explanation if you wish.

.....

Step 1: you need to determine the null and alternative hypotheses. The alternative hypothesis is given to us. Quote: would you agree that the proportion of town voters favoring the proposal is higher than the proportion of county voters? That statement is not one of equality, it does not contain an equal sign or any form of equals to, so it must be the alternative hypothesis. Therefore the alternative hypothesis is that the proportion of town voters favoring the proposal is higher than country voters. From this we can infer the null hypothesis because it is always the opposite: the proportion of town voters favoring the proposal is less than or equal to the proportion of county voters.

H0: the proportion of town voters favoring the proposal is less than or equal to the proportion of county voters favoring the proposal.

HA: the proportion of town voters favoring the proposal is higher than proportion of country voters favoring the proposal.

.......

I am slightly confused by step 2: The level of significance is clearly 0.05, but z tests do not have degrees of freedom. I would guess your teacher copy pastes these 5 steps for every problem and forgot to remove that piece for this one.

.........

Step 3: The appropriate statistical test is the two sample z test of proportions. You will need to google the formula and do the math here as it is difficult to show my work by typing it out. I will tell you however that you should get a z test statistic of about 2.8697. Please contact me if you'd like me to walk you through how to do this part. Please note that how you choose to set up the formula can reverse the valence of your test statistic. If town voters are proportion 1, then the test statistic should come out positive. If town voters are proportion 2 then the test statistic will be the same value except it will be negative. This does not change the outcome except that the critical value must be consistent with how we choose to do this. More on that in the next section.

......

Step 4: We will need to convert alpha (0.05) into a critical value on the z table. Because this is one tailed we do not cut alpha in half. So what we do here is we search the inside of the table for the value of 0.05. We will find that 0.05 is halfway between the z values of 1.64 and 1.65. 0.05 is a common one so I happen to know that in this case we split the difference. Our z critical value is 1.645. It is positive because it is an upper tailed test (assuming we make town voters proportion 1 and county voters proportion 2). Again, this depends on how you set up the formula and which proportion you consider quote: proportion 1.

.......

Step 5: To reject the null for an upper tailed test our test statistic needs to be greater than our critical value. In fact, it is greater. 2.8697 is greater than 1.645. Therefore we reject the null hypothesis. In plain English, this means that there is indeed evidence that a greater proportion of town voters are in favor of the proposal than county voters.

.......

And I think that about covers it. Please let me know if there are any issues and contact me if you need more help. I hope this explanation was helpful and good luck!