$CLT and probability$

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$CLT and probability$

A researcher conducted an experiment to evaluate if the proportion of supporters for the most corrupted party in the country is equal to 40% (null hypothesis). The alternative hypothesis is that the proportion is larger than 40%. In his sample, she observed:

•An average proportion of 48% of supporters
•An standard error ( ˆσ /sqrt n) of 3.1
•A p-value of 0.005 (approximately)

Assume you can use the CLT because the sample size is large enough. Answer true or false and explain your answer.

(a) If the null hypothesis is right, there is a 0.5% chance of observing a value equal to or larger than 48%.
(b) The probability of the null hypothesis given the data is 0.5%.

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Evvee3355

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Daniel90

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It would take about an hour to write a good solution. Your offer is too low!
Evvee3355

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is it better now?

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Daniel90

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It would take about an hour to write a good solution. Your offer is too low!

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(a) TRUE. We would like to test the Null hypothesis:
\[\mu_0 =40\]
Suppose the null hypothesis holds, then $\mu=40$ and $\sigma=3.1.$ Thus
\[P[X\geq 48]=P[\frac{X-40}{3.1}\geq \frac{48-40}{3.1}]=P[Z\geq 2.580]\]
\[=0.00494\approx 0.005=5 \%.\]
Where $Z$ is the standard normal distribution. So if the null hypothesis is correct, then there is about $5 \%$ chance of observing a value equal to or larger than $48 $.

(b) False. The p-value is a number, calculated from a statistical test, that describes how likely you are to have found a particular set of observations if the null hypothesis were true. The given statement is the opposite which may not be true. Indeed the probability of the mean being equal to a given fixed number is zero as the proportion is a continuous random variable between 0 and 100 and the probability the random variable being equal to a given number is always zero.

Erdos

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Erdos

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If possible, set a later deadline for your questions in future.
Evvee3355

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I will. Unfortunately I only discovered this site last night
Evvee3355

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Thank you

$CLT and probability$

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