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[Undergraduate : Introductory Level Statistics - Hypothesis Testing]

There is a general consensus that 80% of all economists believe inflation will increase for 2018. Call this proportion the null hypothesis. However, there is an alternative view that says that less than 80% of them believe inflation will increase for 2018. Call this the alternative hypothesis. Suppose we follow this decision rule: if we randomly poll 100 economists and find that less than 71% of them believe that inflation will increase in 2018, then reject the null hypothesis. Otherwise, we cannot reject the null hypothesis.

(i) Calculate the probability of committing a type I error for the above decision rule.

(ii) By maintaining the above decision rule, calculate the probability of committing a type II error when in fact only 60% of all economists believe inflation will increase for 2018.

I understand the general idea of the question, however, my difficulty lies in comprehending what the errors look like on a standard normal curve. My thinking is that as a result of the Central Limit Theorem, $\hat{p} \sim N (0.8\, , \, 0.04^2)$, and therefore the test statistic can be calculated as -2.25. How do I interpret this test-statistic though in terms of the first question? Is $P(Z<-2.25) = 0.0122$ the required probability? My lecturer said that this was the method but I do not understand what it means when the probability is 0.0122.

With regards to the second part, I am unable to compute this as well. How do I visualise / compute the type II probability on a standard normal curve? Thank you in advance.

Garen
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  • I have not checked the calculations but $0.0122$ is the probability that (assuming $H_0$ is correct) you get fewer than $71$ economists agreeing, and is the probability in the left tail of the distribution centred at $80$. For the second part, you get a different curve centred at $60$ so its left tail below $71$ is much larger and is in fact most of the distribution. – Henry May 19 '21 at 14:20

1 Answers1

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As per the definition that you surely will find in your lectures, Type I error is "the probabilty to reject $H_0$ when is is true", say

$$\mathbb{P}\left[ \text{Reject}H_0|p=0.80 \right]$$

thus

$$0.0122=\mathbb{P}[Z<-2.25]$$

is the error that you will commit in rejecting your null hypothesis under the condition that it is true.

In the same manner, using the definiton, Type II error is the error that you commit if you Accept (better, you do not Reject) the null hypothesis under the condition that the alternative is true, say

$$\mathbb{P}\left[ \hat{p}>0.71|p=0.6 \right]=\mathbb{P}\left[Z>\frac{0.71-0.6}{\sqrt{\frac{0.6\cdot0.4}{100}}} \right]=1-\Phi(2.245)\approx0.0124$$

Visualization

enter image description here

the left picture's purple area is Type I error, $1.22\%$ while the right picture's purple area is Type II error, $1.24\%$

tommik
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