Testing Claims About Proportions
In Exercises 9–32, test...

Question

Testing Claims About Proportions

In Exercises 9–32, test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, P-value, or critical value(s), then state the conclusion about the null hypothesis, as well as the final conclusion that addresses the original claim. Use the P-value method unless your instructor specifies otherwise. Use the normal distribution as an approximation to the binomial distribution, as described in Part 1 of this section.

Internet Use A random sample of 5005 adults in the United States includes 751 who do not use the Internet (based on three Pew Research Center polls). Use a 0.05 significance level to test the claim that the percentage of U.S. adults who do not use the Internet is now less than 48%, which was the percentage in the year 2000. If there appears to be a difference, is it dramatic?

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. A recent survey of 4200 Canadian adults found that 630 do not own a smartphone. In 2012, 35% of Canadian adults did not own a smartphone. At the 0.05 significance level, test the claim that the current percentage of Canadian adults who do not own a smartphone is now less than 35%. If a difference is found, is it substantial? Awesome. So it appears for this particular problem, we're asked to, at this 0.05 significance level, we're asked to test the claim that the current percentage of Canadian adults who do not own a smartphone is now less than 35%, and then we're asked to determine whether or not if a difference is found, is it considered substantial. So now that we know what we're ultimately trying to solve for, let's read off our multiple choice answers to see what our final answer might be. A is substantial difference, B is moderate difference, C is minor difference, and D is not significant. Awesome. So our first step that we need to take in order to solve for this particular problem is we need to define the hypothesis. So let us denote P to be the current proportion of Canadian adults who do not own a smartphone. So that would mean that the null hypothesis, which we'll call H0, that would mean that H0 would be that P. Once again, P is the current like proportion, so. Once again, P is the current proportion of Canadian adults who do not own a smartphone. So that would mean for the null hypothesis that P is greater than or equal to 0.35, which 0.35 is just 35% written as a decimal. And we also need to define our alternative hypothesis, which will denote as H1, and our alternative hypothesis states that P is less than 0.35, and we need to note that we're testing if the percentage is less than 35%. So with that said. We need to note and recall, or rather recognize that this is a left-tailed Z test for proportions. So now our second step is we need to gather our sample data. So we need to write down all of our known variables. So we're told that our sample size N is equal to 4200, and we need to note that the number without a smartphone, which we'll call this value X, is equal to 630. So that would mean that our sample proportion, which is Phat, and Phat is equal to 630 divided by 4200, which will be equal to when we plug that into our calculator, 0.15. So now our third step is we need to test the statistic, and in order to do that, we need to recall and use the Z formula for a population proportion. So we're trying to use the Z formula for a population proportion. So that would mean that Z. As we should recall in the user equation that Z is equal to Phat minus P 0 divided by the square root of P0 multiplied by, so we need to take P0 multiplied by 1 minus P. 0 Divided by N. Fantastic. So once again, to quickly recap here. Z is equal to Phat minus P0, all divided by the square root of P0 multiplied by 1 minus P0 divided by N. And let's take a moment here to note that Phat in this case is going to be equal to 0.15. And let us note that P0 is going to be equal to 0.35, and N is going to be equal to 4200. So now we need to plug in all of our known variables into our equation and solve for Z. So that would mean that Z is going to be equal to 0.15. So 0.15 minus 0.35. And then we're gonna take this and we're gonna divide it. So we're taking 0.15 minus 0.35 divided by The square root of 0.35 multiplied by 1 minus 0.35, all divided by our value for N, which is 4200. And when we plug that into our calculator and we round to two decimal places, we will find that it's approximately equal to negative 27.17, so negative 27.17. And once again, that's when we round to two decimal places. So now, our 4th step. We need to determine the P value and make our final conclusions. So let us note that a Z score of -27.17 is considered extremely low, which I'll call EL for short. So that means that our value of -27.17 is considered an extremely low value, and the P value is nearly 0, well below the 0.05 significance level. So we can reject the null hypothesis. So I'm going to put a big red X next to H0 because H0 denotes our null hypothesis. So that means as a result, as you as we should recall, there is strong evidence that less than 35% of Canadian adults currently do not own a smartphone. So that means that our fifth and final step is we need to consider if this difference is substantial. So let us note that in 2012, the proportion was 35%. So in 2012, the proportion was 35%. The current estimate right now, currently is 15%, so we need to note that this is a drop, so I'll put an arrow pointing down, a drop of 20%. So that will mean that our final answer without a doubt will be yes, the difference is substantial, both statistically and practically. So that means without a doubt, our final answer has to be multiple choice answer letter A, which once again is substantial difference. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Key Concepts

Hypothesis Testing

P-value

Normal Approximation to the Binomial Distribution

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