Yoshida Toimi is a candidate for the mayor of a medium-sized...

Yoshida Toimi is a candidate for the mayor of a medium-sized Midwestern city. If he receives more than $50 \%$ of the votes, he will win the election. Prior to the election, his campaign staff is planning to ask 100 randomly selected voters if they support Yoshida. a. How many positive responses from this sample of 100 is required so that the probability of $50 \%$ or more voters supporting him is 0.95 or more? b. Carefully state the assumptions required for your answer in part (a). c. Suppose the campaign is able to ask 400 randomly selected voters. Now what is your answer to the question in part (a)?

Yoshida Toimi is a candidate for the mayor of a medium-sized Midwestern city. If he receives more than $50 \%$ of the votes, he will win the election. Prior to the election, his campaign staff is planning to ask 100 randomly selected voters if they support Yoshida.
a. How many positive responses from this sample of 100 is required so that the probability of $50 \%$ or more voters supporting him is 0.95 or more?
b. Carefully state the assumptions required for your answer in part (a).
c. Suppose the campaign is able to ask 400 randomly selected voters. Now what is your answer to the question in part (a)?

Key Concepts

Random Sampling and Underlying Assumptions

The validity of inferences made from the sample depends heavily on the assumptions of randomness and independence. The concept assumes that the sample of individuals is representative of the overall population, the responses are independent, and that the process by which the data are collected does not introduce any biases. These assumptions are critical to ensuring that the results and subsequent interpretations, such as the confidence in meeting a particular threshold, are reliable.

Normal Approximation

When the sample size is large, the binomial distribution can be approximated by a normal distribution due to the Central Limit Theorem. This simplification is useful in making inferences about the binomial proportion, setting thresholds, and computing approximate confidence intervals, though one must ensure that conditions for the approximation are sufficiently met.

Binomial Distribution

This concept refers to the probability distribution governing the outcomes of independent yes/no (success/failure) experiments in which the probability of success is constant. It applies when we count the number of successes (for example, positive responses) in a fixed number of trials, and its properties are fundamental to understanding inference problems that involve proportions derived from sample data.

Confidence Intervals for Proportions

In statistical estimation, a confidence interval for a proportion provides a range within which the true population proportion is likely to lie, with a specified level of confidence (such as 95%). In situations where one wants to make a decision about whether a candidate has majority support, determining a one?sided confidence bound can help determine the minimum number of positive responses needed to assert with high certainty that the population proportion exceeds a critical threshold (e.g., 50%).

Transcript

00:01 Well, here in question a, i need to find the probability that the prediction is correct.

00:06 So i have to find the probability that x is at least 13.

00:11 And here is 25, p is .6.

00:17 So i'm using the binomial approximation to normal.

00:22 Now let's find the mean.

00:24 That's 15 and now let's find the standard deviation.

00:35 So this is 2 .45.

00:40 Now i can find this probability and we want to make sure that 13 is included then i'm looking for this probability when i make the adjustment to the normal correction.

00:54 Now let's convert this to that score and now let's use the score table to find the result.

01:10 So that's approximately 0 .840.

01:17 Or point yeah let's just leave it at this point 846 point 846 now let's move on to question b so what's in question b in question b i have to determine the sample size which is needed so that probability of predicting the correct outcome is at least 0 .95 let's do this so basically it means that this probability that x is greater sum x critical which is n over 2 because here we want to to predict the winner and it's better to say than n is actually more than n over 2 because as if it's n over 2 then it's a tie and here there's a probability that that is more than n over 2 minus the mean divided by the standard deviation and here you know probably since we want to be safer that we definitely take into account who is the winner so we don't we don't calculate the tie i'm going to at 0 .5 to n over 2 from this i subtract 15 and i divide this by the standard deviation of 2.

03:30 So this probability is 1 .645, at least 1 is at least 0 .95.

03:37 And it implies that this score must be 1 .645.

03:47 So now let's calculate this.

04:02 And actually here we need to be more careful because now n is different, so the mean is not 15.

04:08 And the standard deviation is also not different.

04:10 So let's now rewrite the adjusted calculation, is adjusted formula...

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