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(Solved): Problem 1: Deriving the Lower-Tailed Hypothesis Test Consider testing the set of hypothesis [H0: ...

$Problem 1: Deriving the Lower-Tailed Hypothesis Test Consider testing the set of hypothesis \[ \left[\begin{array}{ll} H_{0}:$

$I Result: Upper-Tailed Test When testing the hypotheses \[ \left[\begin{array}{ll} H_{0}: & p=p_{0} \\ H_{A}: & p>p_{0} \end{$

Problem 1: Deriving the Lower-Tailed Hypothesis Test Consider testing the set of hypothesis

[H_{0} : H_{A} : p = p_{0} p < p_{0}

at an arbitrary

?

level of significance. Define the test statistic TS to be

TS = \frac{p ? p _{0}}{\frac{p _{0} ( 1 ? p _{0} )}{n}}

a. Show that

TS ?^{H} N (0, 1)

. If your answer depends on a set of conditions to be true, explicitly state those conditions. b. Argue, in words, that the test should be of the form

decision(TS) {reject H_{0} fail to reject H_{0} if TS < c otherwise

for some constant

c

. As a hint, look up the logic we used in Lecture 13 to derive the twotailed test, and think in terms of statements like "

\overset{p}{^}

is far away from

p_{0}

". You do not have to find the value of

c

in this part. c. Now, argue that

c

must be the

?^{th}

percentile of the distribution of the standard normal distribution (not scaled by negative 1), thereby showing that the full test takes the form

decision(TS) = {reject H_{0} fail to reject H_{0} if TS < z_{?} otherwise

where

z_{?}

denotes the

(?) \times 10 0^{th}

percentile of the standard normal distribution. I Result: Upper-Tailed Test When testing the hypotheses

[H_{0} : H_{A} : p = p_{0} p > p_{0}

at an

?

level of significance, the test takes the form

decision(TS) = {reject H_{0} fail to reject H_{0} if TS > z_{1 ? ?} otherwise

where: -

TS = \frac{p ^ ? p _{0}}{\frac{p _{0} ( 1 ? p _{0} )}{n}}

z_{1 ? ?}

denotes the

(1 ? ?) \times 10 0^{th}

percentile of the standard normal distribution. provided that -

n p_{0} ? 10

n (1 ? p_{0}) ? 10

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Sure, I can help you with that. **a. Show that TS^ underline H 9 mathcal N(0, 1) . If your answer depends on a set of conditions to be true, explicitly state those conditions.** The sampling distribution of the test statistic TS is given by: ``` ``` This is a standard normal distribution with mean 0 and standard deviation . The conditions for this to be true are: * The sample size n must be large enough. * The population must be normally distributed. * The sample must be randomly selected from the population. **b. Argue, in words, that the test should be of the form decision(TS) = (reject Ho fail to reject Ho otherwise for some constant c. As a hint, look up the logic we used in Lecture 13 to derive the two- tailed test, and think in terms of statements like "p is far away from p_{0} ^ prime prime . You do not have to find the value of c in this part.** The test should be of the form decision(TS) = (reject Ho fail to reject Ho otherwise if the test statistic TS is far away from the hypothesized value p0. The value of c is the critical value that separates the rejection region from the non-rejection region. The logic for this is similar to the logic used in Lecture 13 to derive the two-tailed test. In that case, we used the fact that the p-value is the probability of obtaining a test statistic at least as extreme as the one observed, under the null hypothesis. We then set a significance level alpha, and rejected the null hypothesis if the p-value was less than alpha. In this case, we are using a one-tailed test, so we only need to consider the probability of obtaining a test statistic that is less than the hypothesized value p0. We can set a significance level alpha, and reject the null hypothesis if the test statistic is less than the critical value c, which is the (1-alpha)th percentile of the standard normal distribution.

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(Solved): Problem 1: Deriving the Lower-Tailed Hypothesis Test Consider testing the set of hypothesis [H0: ...

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