Net worth is defined as total assets (value of house, cars, mo...

Question

Net worth is defined as total assets (value of house, cars, money, etc.) minus total liabilities (mortgage balance, credit card debt, etc.). According to a recent study by TNS Financial Services, 7% of American households had a net worth in excess of $1 million (excluding their primary residence). A random sample of 1000 American households results in 82 having a net worth in excess of $1 million. Explain why the results of this survey do not necessarily imply that the proportion of households with a net worth in excess of $1 million has increased.

Accepted Answer

Hello everyone, let's take a look at this question together. A survey finds that 72% of city residents support a new public transportation initiative. In a random sample of 150 residents, 90 express support for the initiative. After performing a one sample proportion Z test, you calculate a Z score of -2.67. At the alpha equals 0.05 significance level, is this sample representative of the population? Is it answer choice A? Yes, the sample is representative. Answer choice B, no, the sample is not representative. Answer choice C, the sample is biased, or answer choice D, the Z score is not significant. So in order to solve this question, we have to determine whether the sample is representative of the population at the alpha equals 0.05 significance level, given our calculated Z score of -2.67 from a one sample proportion Z test using the data provided from a random sample of 150. Residents where 90 expressed support for the initiative. And so the first step in determining if the sample is representative of the population is to state the hypotheses, starting with our null hypothesis, which is P is equal to 0.72, and our alternative hypothesis, which is P does not equal 0.72. And then we can determine the critical Z value with a significant level of alpha equals 0.05, and so at alpha equals 0.05 for a two-tailed test, the critical values are positive and negative 1.96, and now we are ready to make our decision, which using our given Z score of -2.67, we can compare that Z score to The critical values where we have -2.67 is less than our critical value of -1.96, and since -2.67 is less than -1.96, this means that the Z score falls outside of the range of -1.96 to 1.96, and as a result, we reject the null hypothesis. Therefore, we can conclude that the sample is not representative of the population at the alpha equals 0.05 level. So looking at our answer choices, we can identify that answer choice B is the correct answer, as we know that the sample is not representative of the population, and all other answer choices are incorrect. I hope you found this video to be helpful. Thank you and goodbye.

Key Concepts

Sampling Variability

Hypothesis Testing and Statistical Significance

Confidence Intervals

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