1. In a recent election Smith ran against Jones for Governor. A pre-election poll surveyed 1885 randomly selected voters. Of those surveyed 980 indicated that they planned to vote for Smith. The five questions below pertain to the problem of testing the hypothesis that a majority of voters favor Smith. Testing is to be done at the 0.01 significance level. First find the sample proportion of voters who plan to vote for Smith.
Answer:
2. Find the sample standard deviation.
Answer:
3. Find the test statistic.
Answer:
4. Find the P-value.
Answer:
5. Is the claim that a majority of the voters favor Smith supported at the 0.01 significance level? Enter 1 for yes, 0 for no.
Answer:
Here's a few hints.
1. To find the sample proportion of voters, use x/n. x = number who planned to vote for Smith. n = sample size. You can convert this to a decimal if asked to do so.
2. Sample standard deviation = √npq
n = sample size. p & q both equal .5 if no value is stated.
3. Use a z-test formula. You will need a population proportion or a population mean, which I don't see stated in the problem.
4. The p-value is the actual level of the test statistic, which can be found using a z-table.
5. If the p-value is greater than .01, the claim cannot be supported. If the p-value is less than .01, then the claim can be supported.
I hope this will help.
To answer these questions, we need to follow a step-by-step process. Here's how you can find the answers:
1. Find the sample proportion of voters who plan to vote for Smith:
The sample proportion, denoted as p̂ (pronounced "p-hat"), is calculated by dividing the number of voters who plan to vote for Smith (980) by the total number of voters surveyed (1885):
p̂ = 980/1885 ≈ 0.519
2. Find the sample standard deviation:
The sample standard deviation, denoted as s, measures the spread of the data. To calculate it, first find the sample variance (s^2) by using the formula:
s^2 = (p̂(1 - p̂))/n
where n is the number of observations (1885 in this case). Then take the square root to find the standard deviation:
s = √(s^2)
3. Find the test statistic:
In this case, since the sample size is large (n > 30), we can use the z-test statistic. The formula for the z-test statistic is:
z = (p̂ - p0)/√((p0(1 - p0))/n)
where p0 is the hypothesized population proportion. In this case, we are testing the hypothesis that a majority of voters favor Smith, so p0 = 0.5.
4. Find the P-value:
The P-value is the probability of observing a sample proportion as extreme, or more extreme, than the one obtained, given that the null hypothesis (no majority in favor of Smith) is true. To find the P-value, we need to compare the absolute value of the z-test statistic to the critical value(s) from the standard normal distribution table.
5. Determine if the claim is supported:
Compare the P-value to the significance level (0.01 in this case). If the P-value is less than the significance level, we reject the null hypothesis and conclude there is evidence to support the claim.
Note: I provided the steps involved in finding the answers, but I cannot calculate the specific values without the actual data.