Let p be the (unknown) true fraction of voters who support a particular candidate A for office.

6. Let p be the (unknown) true fraction of voters who support a particular candidate A for office. To estimate p, we poll a random sample of n voters. Let Fn be the fraction of voters who support A among n randomly selected voters.

A. Using Chebyshev’s Inequality, calculate an upper bound on the probability that if we poll 100 voters, our estimate Fn differs from p by more than 0.1. (Hint: you may need to use the fact that x(1 − x) ≤ 1/4 for any 0 ≤ x ≤ 1).

B. How many voters need to be polled if we want to have high confidence (probability at least 95%) that our estimate differs from p by at most 0.01? Use Chebyshev’s Inequality.

7. Repeat question 6 using the Central Limit Theorem to calculate probabilities

Solutions

Expert Solution

orchestra answered 1 year ago

Next > < Previous

Related Solutions

Suppose that 60% of registered voters support candidate A. A pollster will obtain a random sample...

Suppose that 60% of registered voters support candidate A. A pollster will obtain a random sample of 200 registered voters. What is the probability that at least 126 of the individuals in the sample support A? Do not employ a continuity correction. 0.8078 0.2447 0.1922 0.0749 0.5956

You are conducting a study to see if the proportion of voters who prefer Candidate A...

You are conducting a study to see if the proportion of voters who prefer Candidate A is significantly more than 47%. With Ha : p > 47% you obtain a test statistic of z = 1.997. Find the p-value accurate to 4 decimal places. p-value = You are conducting a study to see if the accuracy rate for fingerprint identification is significantly less than 0.62. Thus you are performing a left-tailed test. Your sample data produce the test statistic z...

You are conducting a study to see if the proportion of voters who prefer Candidate A...

You are conducting a study to see if the proportion of voters who prefer Candidate A is significantly more than 0.82. You use a significance level of α=0.10α=0.10. H0:p=0.82H0:p=0.82 H1:p>0.82H1:p>0.82 You obtain a sample of size n=202n=202 in which there are 181 successes. What is the p-value for this sample? (Report answer accurate to four decimal places.) p-value =

You are conducting a study to see if the proportion of voters who prefer Candidate A...

You are conducting a study to see if the proportion of voters who prefer Candidate A is significantly different from 66% at a significance level of αα = 0.10. According to your sample, 60 out of 96 potential voters prefer Candidate A. For this study, we should use Select an answerχ²-Testχ²GOF-Test2-PropZInt2-SampTInt1-PropZInt2-PropZTestANOVA2-SampTTestT-TestTInterval1-PropZTest The null and alternative hypotheses would be: H0H0: ?μp ?≠><= (please enter a decimal) H1H1: ?μp ?<=>≠ (Please enter a decimal) The test statistic = (please...

Suppose that a particular candidate for public office is in fact favored by 49% of all...

Suppose that a particular candidate for public office is in fact favored by 49% of all registered voters in the district. A polling organization will take a random sample of 550 voters and will use p̂, the sample proportion, to estimate p. What is the approximate probability that p̂ will be greater than 0.5, causing the polling organization to incorrectly predict the result of the upcoming election? (Round your answer to four decimal places.)

A survey is conducted using 2000 registered voters who are asked to choose between candidate A...

A survey is conducted using 2000 registered voters who are asked to choose between candidate A and candidate B. Let p denote the fraction of voters in the population who prefer candidate A and let ˆp denote the fraction of voters who prefer Candidate B. i You are interested in performing the following hypothesis test H0 : p = 0.4 (1) H1 : p 6= 0.4 (2) Determine the size of the test and compute the power of the test...

Q1. After surveying 625 potential voters polled, you find out that 325 support candidate X and...

Q1. After surveying 625 potential voters polled, you find out that 325 support candidate X and 300 support candidate Y. a. What is the 95% confidence interval for the support rate for X? b. Based on your survey, what is the chance that the candidate Y will WIN in a general election ? c. If you want to shrink the standard deviation of your survey to 1.5%, how many people do you need to survey? Please show work for all

In the next U.S Presidential Election, would you be more likely to support a candidate who...

In the next U.S Presidential Election, would you be more likely to support a candidate who favors international trade, or a candidate who opposes international trade? Why? What arguments could that candidate make that might persusde their opponents to agree with them?

Let p be the (unknown) proportion of males in a town of 100, 000 residents. A...

Let p be the (unknown) proportion of males in a town of 100, 000 residents. A political scientist takes a simple random sample of 100 residents from this town. (a) Write down the exact pmf, as well as an approximate pmf, for the number of males in the sample. (They should both depend on p). (b) If the number of males in the sample is 55 or more, the political scientist will claim that there are more males than females...

Let X~Geometric(p), with parameter p unknown, 0<p<1. a) Find I(p), the Fisher Information in X about...

Let X~Geometric(p), with parameter p unknown, 0<p<1. a) Find I(p), the Fisher Information in X about p. b) Suppose that pˆ is some unbiased estimator of p. Determine the Cramér-Rao Lower Bound for Var p[ ]ˆ based on one observation from this distribution. c) Show that p= I {1}(X) is an unbiased estimator of p. Does its variance achieve the Cramer-Rao Lower Bound?

Subjects