A sample of 100 students is to be taken to determine which...

A sample of 100 students is to be taken to determine which of two brands of beer is preferred in a blind taste test. Suppose that, in the whole population of students, $50 \%$ would prefer brand $\mathrm{A}$. a. What is the probability that more than $60 \%$ of the sample members prefer brand A? b. What is the probability that between $45 \%$ and $55 \%$ of the sample members prefer brand A? c. Suppose that a sample of only 10 students was available. Indicate how the method of calculation of probabilities would differ, compared with your solutions to parts (a) and (b)?

A sample of 100 students is to be taken to determine which of two brands of beer is preferred in a blind taste test. Suppose that, in the whole population of students, $50 \%$ would prefer brand $\mathrm{A}$.
a. What is the probability that more than $60 \%$ of the sample members prefer brand A?
b. What is the probability that between $45 \%$ and $55 \%$ of the sample members prefer brand A?
c. Suppose that a sample of only 10 students was available. Indicate how the method of calculation of probabilities would differ, compared with your solutions to parts (a) and (b)?

Key Concepts

Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent trials, where each trial has two possible outcomes and a constant probability of success. In the context of preference testing or surveys, it is used to calculate the likelihood of a certain number of individuals favoring one option over another.

Normal Approximation and the Central Limit Theorem

When the sample size is large, the binomial distribution can be approximated by a normal distribution due to the central limit theorem. This theorem states that the distribution of the sample proportion will approach a normal distribution as the sample size increases, which simplifies the calculation of probabilities related to proportions.

Continuity Correction

When using a continuous distribution, such as the normal approximation, to estimate probabilities for a discrete distribution like the binomial, the continuity correction is applied. This involves adjusting the values by 0.5 to account for the discreteness of the actual probability mass function, thereby improving the accuracy of the approximation.

Exact Probability Calculation for Small Samples

For small sample sizes, the normal approximation is not reliable, and exact probability calculations must be performed directly using the binomial formula. This method involves summing the probabilities for the specific outcomes of interest, ensuring precise results when the number of trials is too few for the central limit theorem to hold effectively.

Transcript

0:00 Hello.

00:01 So here we found the proportion of the students who preferred beer brand a in the blind test with 0.5.

00:07 So we have that p is going to be equal to 0.5.

00:11 And then in part they were considering a random sample of 100 students from this college.

00:16 Um and we're interested in computing the probability that more than 60% of the students from this sample preferred brand eight.

00:24 So to calculate the probability that p hat is greater than 0.6.

00:28 So we have the sample proportion is p hat has a normal distribution with mean mu is equal to p which is equal to 0.5.

00:36 Therefore the standard deviation sigma is going to be equal to the square root of p times one minus p.

00:43 All over ends, just to square root of 0.5 times 0.5.

00:47 All over 100 which is going to be equal to 0.5.

00:51 Um and then for our probability that's going to be equal to one minus the probability that z is less than equal to 0.6 minus 0.5 are divided by 0.5 is going to be equal to one minus the probability that z is less than equal to two, which is going to be equal to zero 0.228.

01:15 So we have that the probability that more than 60% of the students from this sample have preferred on the brand is going to be equal to 0.0 2 to 8.

01:28 And then in part b we want to compute the probability that were between 45% and 60%...

Please give Ace some feedback