In Exercises 3–6, determine whether a normal sampling distribu...

Question

In Exercises 3–6, determine whether a normal sampling distribution can be used. If it can be used, test the claim.

Claim: p <0.12, α=0.01. Sample statistics: p_hat = 0.10, n=40

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. In a study of users adoption of a new app feature, researchers claim that the true adoption rate P is greater than 30%. They collect a sample of N equals 100 users and find a sample proportion Phat. is equal to 0.35 alpha equals 0.05, can they use the normal approximation and do the data support the claim? Awesome. So it appears for this particular prong we're asked to take a significance level of LF equals 0.05, and we're asked if they can use the normal approximation and does this data support the claim. So that's our final answer that we're ultimately trying to solve for. So based on the information that is provided to us by the prom itself, can they use the normal approximation and does the data support their claim? Where the claim is that the true adoption rate P is greater than 30%. So with that in mind, let's read off our multiple choice answers to see what our final answer might be. A is the normal approximation can be used, but the data do not support the claim that the adoption rate exceeds 30% at alpha equals 0.05. B is the normal approximation cannot be used, and the data support the claim that the adoption rate exceeds 30% at alpha equals 0.05. C is the normal approximation can be used, and the data support the claim that the adoption rate exceeds 30% at alpha equals 0.05. And D is the normal approximation cannot be used, and the data do not support the claim that the adoption rate exceeds 30% at alpha equals 0.05. Awesome. So first off, in order to solve this particular problem, we need to check the normal approximation. So as we should recall, N multiplied by P0 is equal to 100 multiplied by 0.30, which is equal to 30, and N multiplied by 1 minus P0 is equal to 100 multiplied by 0.70, which is equal to 70, and both of these values, which I'll just call B for short, are greater than or equal to 10. So we can use the normal approximation. So now we need to determine what the null and alternative hypotheses are for this particular prompt. So the null hypothesis H0 states that P is equal to 0.30, and the alternative hypothesis HA states that P is greater than 0.30, which we need to note and recall, or rather recognize that this is a right-tailed test, which we'll call it RT for short. So this is a right-tailed test. So with that in mind, we now need to determine the test statistic Z. So as we should recall, Z is equal to Phat minus P0 divided by the square root of P0 multiplied by 1 minus P0, all divided by N, which plugging in our non variables, we will find that Z is equal to Our value for Phat, which is 0.35 minus P0, which is 0.30. Once again we're plugging in all of our known variables to solve for the numerical value of Z divided by the square root of 0.30 multiplied by 0.70. All divided by our value of N, which our value for N is 100. So that will mean when we plug this into our calculator, when we plug our expression into our calculator, we will find that Z is approximately equal to, when we round to two decimal places, 1.09. Fantastic. So now we need to solve for the critical value, so we know that we're dealing with a right tailed at alpha equals 0.05. And that it's right tailed, so it's R dash T. So that will mean that our critical value Z subscript 0.95 is equal to 1.645. Fantastic. So now we need to make our decision. So since 1.09 is less than 1.645, we fail to reject the null hypothesis, so I'll put an X next to H0 which denotes the null hypothesis. So because 1. 9 is less than 1.645. We fail to reject the null hypothesis, which means there is insufficient evidence at the 5% level to conclude that the adoption rate exceeds 30%. So that means our final answer, without a doubt, will have to be multiple choice answer, letter A, which once again states the normal approximation can be used, but the data do not support the claim that the adoption rate exceeds 30% at alpha equals 0.05. And that's it. That is our final answer. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

In Exercises 3–6, determine whether a normal sampling distribution can be used. If it can be used, test the claim.
Claim: p <0.12, α=0.01. Sample statistics: p_hat = 0.10, n=40

Key Concepts

Normal Sampling Distribution

Hypothesis Testing

Significance Level (α)

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