In Problems 1–6, test the hypothesis using the P-value approac...

Question

In Problems 1–6, test the hypothesis using the P-value approach. Be sure to verify the requirements of the test.

H₀: p = 0.3 versus H₁: p > 0.3

n = 200; x = 75; α = 0.05

Accepted Answer

Hello. In this video, we are going to be using our hypothesis testing methods. So we are told that to test the claim that the proportion of twos is 0.1 at the 0.05 significance level, using the P-value method, we want to consider the following scenario. In a survey of 800 randomly chosen credit card numbers, 68 of the last digits were 2s. While if the digits were truly random, the proportion of twos should be 0.1. OK, so let's go ahead and start this problem by setting up our null hypothesis and our alternate hypothesis. Now we are told that we want. The proportion of 2s to be 0.1 at a 0.05 significance level. So because the 0.1 is our target value, that is going to be our null hypothesis. So our null hypothesis, H subzero is such that P is equal to 0.1. Now we need to set up our alternate hypothesis, and the alternate hypothesis focuses on the opposite ends. The alternate hypothesis is the claim. That the proportion of twos is not equal to 0.1. So this is going to be our no hypothesis and alternate hypothesis for the hypothesis testing. Furthermore, because we are testing the differences between two values, we want to use a two-tailed test. So, let's go ahead and start by calculating our sample proportion. Now, in order to calculate the sample proportion, there are 2 things that we need. The first thing we need is we need our sample size, and we need to figure out how many people Have the last digits that were twos. Now, in the problem, we are told that 800 people were randomly chosen. So we have a sample size N equal to 800. Furthermore, we are told that we have 68 people whose last digits of the cards were twos. So we have that X, or which is our population of interest, is equal to 68. Now, in order to calculate the sample proportion. Which is going to be Phat. Phat is defined as the population of interest divided by our sample size N. This is going to be 68 divided by 800, which will give us the value of 0.085. So this is going to be our sample proportion. Now what we want to do is want to go ahead and calculate our Z score. We are going to use a Z score because we have a sample size that is larger than 30. So, the Z score is defined as the following. It is our sample proportion minus. Our initial proportion. Divided by the square root of our sample proportion multiplied by its complement. Divided by our sample size n. Now, again, our main proportion P knot is the value from the null hypothesis, 0.1. So if we take our information, plug in Phat, our Pannot, and our sample size, that is going to give us a Z score of 0.085 minus 0.1, all divided by the square root. Of 0.085, multiplied by 1 minus 0.085, all divided by 800. Now, if we simplify this with a calculator, in the numerator, we get 0.015, and in the denominator, we get 0.0106. And simplifying this further will give us an approximate value of -1.414, which is going to be our Z score. Now, what we need to do is we need to find our P-value. Now, in order to find the p-value, recall that we are using a two-tailed test. What we want to do is we want to calculate the areas on the tail ends of our normal distribution. So we want to find the probability that Z is less than -1.414. Now, by using a T Z table, By using the Z table, we get that this probability is equal to 0.0787. Now keep in mind that this is the area under one tail end of the normal distribution, but we need the area underneath both. So now what we need to do is we need to multiply this value by 2 to consider the area under the right tail end of the normal distribution. And this is going to give us the value of 0.1574. Which is going to be our P value. Now that we have our P value, we want to compare this P-value to our significance level. Now, recall that our significance level. It is 0.05. It was given to us in the beginning of the problem. But notice that our P value is 0.1574. This is greater than 0.05, and since our P-value is greater than our significance level, we fail. To reject Our null hypothesis. And what this suggests is that this is not sufficient evidence to suggest that the proportion of twos at the end of cards is different from 0.1. And with that being said, the solution to this problem is going to be C. So I hope this video helps you in understanding no hypothesis testing, and we will go ahead and see you all in the next video.

In Problems 1–6, test the hypothesis using the P-value approach. Be sure to verify the requirements of the test.

H₀: p = 0.3 versus H₁: p > 0.3
n = 200; x = 75; α = 0.05

Key Concepts

Hypothesis Testing

P-value Approach

Requirements for Testing a Population Proportion

Watch next