A random sample of 415 potential voters was interviewed 3 weeks before the start of a state-wide campaign for governor; 223 of the 415 said they favored the new candidate over the incumbent. However, the new candidate made several unfortunate remarks one week before the election. Subsequently, a new random sample of 630 potential voters showed that 317 voters favored the new candidate. Do these data support the conclusion that there was a decrease in voter support for the new candidate after the unfortunate remarks were made?

Yes. There is  enough evidence to support the claim that  there was a decrease in voter support for the new candidate after the unfortunate remarks.

Step-by-step explanation:

This is a hypothesis test for a proportion.

The previous proportion, to which we will test the new proportion, was:

The claim is that  there was a decrease in voter support for the new candidate after the unfortunate remarks.

Then, the null and alternative hypothesis are:

The significance level is 0.05.

The sample has a size n=630.

The sample proportion of the new sample is p=0.5032.

The standard error of the proportion is:

Then, we can calculate the z-statistic as:

This test is a left-tailed test, so the P-value for this test is calculated as:

As the P-value (0.0468) is smaller than the significance level (0.05), the effect is  significant.

The null hypothesis is rejected.

There is  enough evidence to support the claim that  there was a decrease in voter support for the new candidate after the unfortunate remarks.

Since is a left tailed test the p value would be:    

For this case if we use a significance level of 10% we see that so then we don;'t have enough evidence to conclude that there was a decrease in voter support for the new candidate after the unfortunate remarks were made

Step-by-step explanation:

Data given and notation    

represent the number of people who support the candidate before

represent the number of people who support the candidate after

sample before

sample after

represent the proportion before

represent the proportion after

represent the pooled estimate of p

z would represent the statistic (variable of interest)    

represent the value for the test (variable of interest)  

significance level given  

Concepts and formulas to use    

We need to conduct a hypothesis in order to check if is the proportion after is lower than the proportion before, the system of hypothesis would be:    

Null hypothesis:    

Alternative hypothesis:    

We need to apply a z test to compare proportions, and the statistic is given by:    

  (1)  

Where  

Calculate the statistic  

Replacing in formula (1) the values obtained we got this:    

Statistical decision  

Since is a left tailed test the p value would be:    

For this case if we use a significance level of 10% we see that so then we don;'t have enough evidence to conclude that there was a decrease in voter support for the new candidate after the unfortunate remarks were made

a.   since the profit made in 2012 is equal to the profit made in 2014, the profit is symmetric about the line y = 120,000.

step-by-step explanation: