Given a random sample size of n=1,600 from a binomial...

Given a random sample size of $n=1,600$ from a binomial probability distribution with $P=0.40$ do the following: a. Find the probability that the percentage of successes is greater than $0.45$. b. Find the probability that the percentage of successes is less than $0.35$. c. Find the probability that the percentage of successes is between $0.37$ and $0.44$. d. With probability $0.20$, the percentage of successes is less than what percent? e. With probability $0.09$, the percentage of successes is greater than what percent?

Given a random sample size of $n=1,600$ from a binomial probability distribution with $P=0.40$ do the following:
a. Find the probability that the percentage of successes is greater than $0.45$.
b. Find the probability that the percentage of successes is less than $0.35$.
c. Find the probability that the percentage of successes is between $0.37$ and $0.44$.
d. With probability $0.20$, the percentage of successes is less than what percent?
e. With probability $0.09$, the percentage of successes is greater than what percent?

Key Concepts

Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success. It is integral for determining probabilities in experiments where outcomes are binary, such as success or failure over a set number of trials.

Normal Approximation

For large sample sizes, the binomial distribution can be approximated by a normal distribution via the Central Limit Theorem. This approximation simplifies calculations by allowing the use of continuous probability techniques in place of discrete ones.

Z-Score

A z-score indicates the number of standard deviations a particular value is from the mean of a distribution. It is used in the context of the standard normal distribution to compute probabilities and to standardize binomial outcomes when applying the normal approximation.

Quantile Calculation

Quantile calculations involve determining the value below or above which a certain proportion of the data falls. This concept is crucial for answering probability questions that ask for thresholds corresponding to specific cumulative probabilities.

Transcript

00:01 Now, here on this problem, we're told that we've got a binomial distribution.

00:04 So the best thing here, whenever we've got large values of the binomial.

00:07 Find a binomial calculator that you can use.

00:09 I use the one here on statcrunch.

00:11 We can use anyone that's publicly available for you.

00:14 Now, we're told that a random sample of 1600.

00:17 So my end value is going to change here to 1600.

00:22 And p is 0 .40.

00:25 Let's make p .0 .40.

00:28 We first want to find the probability that the percentage of successes is greater than 0 .10.

00:33 0 .45.

00:35 Now, 0 .45 times 1 ,600 is 720.

00:39 0 .45 times 1 ,600 is 720.

00:42 And so what this means is that we want to find the probability that the percentage of successes, or the number of successes, is greater than 720.

00:52 So we find the probability x is greater than 720, and this gives us a very, very small number point, 0 -000 -2 -0 -0 -6.

01:02 That's our probability here.

01:04 B, we want to find the probability that the percentage of successes is less than 0 .35.

01:11 Now, 35 % of 1 ,600 is 560.

01:18 So now we're wanting the probability of x is less than 560.

01:25 And this probability computes to a very small number.

01:27 0 .0001744.

01:33 Now, c, we want it to be between 0 .37 and 0 .44.

01:39 Between 0 .37 and 0 .44.

01:43 Now, 0 .37 times 1 ,600, is 592.

01:54 0 .44 times 1 ,600, is 704.

01:58 So this is just asking us, what's the probability it's between 592 and 704, and that gives this value of 0 .9930 when it rounds up.