The probability of someone voting for a particular candidate in a two-person election is 45 % 45\% . Use a normal distribution to approximate the probability that between 62 62 and 70 70 people out of a sample of 100 100 vote for the candidate.

this problem, we're told that the probability of someone voting for a particular candidate in a 2 person election is 45%. We're asked here to use a normal distribution to approximate the probability that between 62 70 people out of a sample of 100 vote for that particular candidate. Now, in this particular problem, we are told explicitly to use a normal distribution to approximate this probability, but remember that in order to be able to do this, both np and nq must be greater than or equal to 5. So we want to double check that here so that we know that we can use our normal distribution. Now in this problem, we have an n value of 100. That is our total sample size. And then p, the probability of success, in this case, voting for that particular candidate, is 45% or 0.45. Now 0.45 times 100 is definitely greater than or equal to 5, and point 55, that's q times 100, is also greater than or equal to 5. So we know that we're good here to continue on approximating using our normal distribution. Now in this problem, we are asked to find the probability that between 6270 people voted for this particular candidate. So you wanna find here the probability that x is greater than 62 and less than 70. Now remember that we need to perform a continuity correction here to account for the fact that our normal distribution is continuous, and we are using it to approximate a discrete binomial distribution. So we need to go ahead and subtract 0.5 from that 62 and add 0.5 to that 70. So we want to find here the probability that x is greater than 61.5 and less than 70.5. So this is the actual probability that we want to find having done our continuity correction. Now whenever we're thinking about finding this probability, thinking about what our normal distribution is, because we're trying to find the probability in between these two values, if we think about these two values of being here, we can just call this x1 and x2. Looking at these or looking at these values on our graph, if we want to find the probability that x is in between these two values, we need to find this large area going all the way up to that second value and our smaller area just going up to that first value. Because if we subtract that smaller area from that larger area, that will give us the area in between and thus our probability that x is in between those two values. So we need to find 2 z scores here in order to use our normal distribution to do this approximation. We need to find the z score associated with this x value of 61.5 and with 70.5. So starting with that 61.5, we're going to refer to this as z one. Remember that in order to find a z score, we need to take x subtract np, divide that by the square root of n pq. That's based on what the mean and standard deviation of our binomial variable are. Now here we're plugging in a 61.5 for x, so 61.5 minus np. So that's 100 times 0.45 based on our problem divided by the square root of npq. So that's 100 times 0.45 times 0.55 because that's 1 minus p. Now doing this algebra here gives us a z score of 3.312, but that's only half the story because we need to find our other z value associated with x being 70.5. So we're going to call this z 2. So plugging in 70.5, doing all the same other values here. So 70.5 minuteus NP. So that's 100 times 0.45. Again, dividing that by the square root of 100 times 0.45 times 0.55. Doing this algebra results in a z score here of 5.126. So based on the original probability that we stated up here, the probability that x is greater than 61.5 and less than 70.5, this is the probability that z is greater than 3.312 and less than 5.126 based on these z scores that we found associated with those x values. Now remember that in order to find this probability, we're gonna have to take this larger area and subtract our smaller area to get this area in between these two values and the probability that we're looking for. So based on that, this probability is going to be equal to the probability that z is less than 5.126, that's our larger area, minus the probability that z is less than 3.312. That is our smaller area. So even though here we're looking at the probability that z is greater than 3.312 because of the subtraction that we have to do, this probability is that z is less than 3.312 just based on the area that we are looking at. Now if we go ahead and use a z table or our calculator here to perform this calculation, we should end up getting that this probability is 0.000463. This is a rather small probability that between 62 and 70 people voted for this particular candidate. Now feel free to let us know in the comments if you have any questions, and we're gonna continue getting practice coming up next.

8. Sampling Distributions & Confidence Intervals: Proportion

Sampling Distribution of Sample Proportion

Statistics

8. Sampling Distributions & Confidence Intervals: Proportion

Sampling Distribution of Sample Proportion

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Multiple Choice

The probability of someone voting for a particular candidate in a two-person election is $45 percent sign$ . Use a normal distribution to approximate the probability that between $62$ and $70$ people out of a sample of $100$ vote for the candidate.

0.00165

0.000463

0.99

0.01

Verified step by step guidance

Identify the problem as a binomial distribution problem where the probability of success (voting for the candidate) is 0.45, and the number of trials (sample size) is 100.

Use the normal approximation to the binomial distribution. First, calculate the mean (μ) and standard deviation (σ) of the binomial distribution. The mean is given by μ = np, where n is the number of trials and p is the probability of success. The standard deviation is given by σ = sqrt(np(1-p)).

Convert the binomial problem to a normal distribution problem by using the continuity correction. Since we want the probability that between 62 and 70 people vote for the candidate, adjust the interval to 61.5 to 70.5 to account for the continuity correction.

Standardize the values using the z-score formula: z = (X - μ) / σ, where X is the value you are standardizing. Calculate the z-scores for 61.5 and 70.5 using the mean and standard deviation calculated earlier.

Use the standard normal distribution table or a calculator to find the probabilities corresponding to the calculated z-scores. Subtract the smaller probability from the larger probability to find the probability that between 62 and 70 people vote for the candidate.