MHB Probability of the error of type 2

AI Thread Summary
The discussion focuses on calculating the probability of a Type II error in a hypothesis test where the null hypothesis is H0: μ ≤ 100 and the alternative hypothesis is H1: μ > 100, with a sample mean of 102 from a population with a standard deviation of 20. The critical sample mean value is determined to be 102.56, leading to the conclusion that if the sample mean is greater than this value, the null hypothesis is rejected. The probability of making a Type II error, which occurs when the null hypothesis is incorrectly accepted despite the true mean being 102, is calculated as 0.6103. Participants clarify the definitions of Type II error and the conditions under which the null hypothesis is retained or rejected. The conversation concludes with a confirmation of the calculations and understanding of the concepts involved.
mathmari
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Hey! :o

We have data of a sample of $100$ people from a population with standard deviation $\sigma=20$.

We consider the following test: \begin{align*}H_0 : \ \mu\leq 100 \\ H_1 : \ \mu>100\end{align*}

The real mean is $\mu=102$ and the significance level is $\alpha=0.1$.

I want to calculate the probability of the error of type 2. I have done the following:

The statistic function is: \begin{equation*}Z=\frac{\overline{X}-\mu}{\sigma_{\overline{X}}}=\frac{\overline{X}-\mu}{\frac{\sigma}{\sqrt{n}}}=\frac{\overline{X}-100}{\frac{20}{\sqrt{100}}}=\frac{\overline{X}-100}{\frac{20}{10}}=\frac{\overline{X}-100}{2}\end{equation*}
where $\overline{X}$ is the estimation of $\mu$.

For the significance level $\alpha=0.1$ the critical value is $Z_{c} = 1.28$ and the region of rejection of $Η_0$ is $R = \{Z\mid Z > 1.28\}$.

The critical value $Z_{c}$ corresponds to a critical value $\overline{X}_{c}$ such that \begin{equation*}P\left (Z>1.28\right )=P\left (\overline{X}>\overline{X}_c \mid \mu=100 , \sigma_{\overline{X}}=2\right ) =1-\alpha=0.9\end{equation*}

We can find the value of $\overline{X}_c$ solving th following equation: \begin{equation*}Z_c=1.28 \Rightarrow \frac{\overline{X}_c-100}{2}=1.28 \Rightarrow \overline{X}_c-100=2.56 \Rightarrow \overline{X}_c=102.56\end{equation*}

So incorrectly we fail to reject the null hypothesis if we take a sample mean greater than $102.56$.

The probability to take a sample mean greater than $102.56$ given $\mu=102$ and $\sigma_{\overline{X}}=2$, i.e. the probability of error of type II is \begin{align*}P\left (\overline{X}>102.56\mid \mu=102, \sigma_{\overline{X}}=2\right )&=P\left (Z>\frac{102.56-102}{2}\right )=P\left (Z>\frac{0.56}{2}\right )=P\left (Z>0.28\right )\\ & =1-P\left (Z\leq 0.28\right )=1-0.6103=0.3897\end{align*} Is everything correct? (Wondering)
 
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mathmari said:
We can find the value of $\overline{X}_c$ solving th following equation: \begin{equation*}Z_c=1.28 \Rightarrow \frac{\overline{X}_c-100}{2}=1.28 \Rightarrow \overline{X}_c-100=2.56 \Rightarrow \overline{X}_c=102.56\end{equation*}

So incorrectly we fail to reject the null hypothesis if we take a sample mean greater than $102.56$.

Hey mathmari!

If we fail to reject the null hypothesis, we keep the null hypothesis don't we?
Isn't that the case if we find a sample mean less than $102.56$? (Wondering)

mathmari said:
The probability to take a sample mean greater than $102.56$ given $\mu=102$ and $\sigma_{\overline{X}}=2$, i.e. the probability of error of type II is \begin{align*}P\left (\overline{X}>102.56\mid \mu=102, \sigma_{\overline{X}}=2\right )&=P\left (Z>\frac{102.56-102}{2}\right )=P\left (Z>\frac{0.56}{2}\right )=P\left (Z>0.28\right )\\ & =1-P\left (Z\leq 0.28\right )=1-0.6103=0.3897\end{align*}

Is everything correct?

I believe you have calculated the so called Power instead of the Type II error. (Worried)
 
Klaas van Aarsen said:
If we fail to reject the null hypothesis, we keep the null hypothesis don't we?
Isn't that the case if we find a sample mean less than $102.56$? (Wondering)

The error of type II is to accept the null hypothesis although it is wrong, isn't it?

I got stuck right now. We found the critival $\overline{X}$-value to be $102.56$ which is greater than $100$ and so we would accept the hypothesis $H_1$, or not? (Wondering)
 
mathmari said:
The error of type II is to accept the null hypothesis although it is wrong, isn't it?

Correct. (Nod)

mathmari said:
I got stuck right now. We found the critival $\overline{X}$-value to be $102.56$ which is greater than $100$ and so we would accept the hypothesis $H_1$, or not?

Not quite.

Let's denote the critical $\overline{X}$-value as $\overline{X_0}^c$ to avoid confusion.
Note that $\overline{X_0}^c$ is calculated based on the null hypothesis for a certain significance level and sample size.
Now if we take a sample $x$ and its mean $\overline x$ is greater than $\overline{X_0}^c$, then we accept the alternative hypothesis, don't we?
And if $\overline x$ is less than $\overline{X_0}^c$, then we keep the null hypothesis, don't we? (Thinking)

The Type II Error is the probability that a sample follows the alternative distribution, but has a mean so close to the null hypothesis that we keep the null hypothesis even though it is wrong.
In our case:
$$\beta = \text{Type II Error} = P(\overline X < \overline{X_0}^c \mid \mu = \mu_1 = 102)$$
where $\overline{X_0}^c$ is calculated based on the null hypothesis with $\mu=\mu_0=100$. (Thinking)
 
Klaas van Aarsen said:
Let's denote the critical $\overline{X}$-value as $\overline{X}^c$ to avoid confusion.
Now if we take a sample $x$ and its mean $\overline x$ is greater than $\overline{X}^c$, then we accept the alternative hypothesis, don't we?
And if $\overline x$ is less than $\overline{X}^c$, then we keep the null hypothesis, don't we? (Thinking)

The Type II Error is the probability that a sample follows the alternative distribution, but has a mean so close to the null hypothesis that we keep the null hypothesis even though it is wrong.
In our case:
$$\beta = \text{Type II Error} = P(\overline X < \overline X^c \mid \mu = \mu_1 = 102)$$
where $\overline X^c$ is calculated based on the null hypothesis with $\mu=\mu_0=100$. (Thinking)
Ah ok! So do we have the following? (Wondering)

\begin{equation*}\beta = P(\overline X < \overline X^c \mid \mu = \mu_1 = 102)=P\left (Z<\frac{102.56-102}{2}\right )=P\left (Z<\frac{0.56}{2}\right )=P\left (Z<0.28\right )=0.6103\end{equation*}
 
mathmari said:
Ah ok! So do we have the following?

\begin{equation*}\beta = P(\overline X < \overline X^c \mid \mu = \mu_1 = 102)=P\left (Z<\frac{102.56-102}{2}\right )=P\left (Z<\frac{0.56}{2}\right )=P\left (Z<0.28\right )=0.6103\end{equation*}

Yep. (Nod)
 
Klaas van Aarsen said:
Yep. (Nod)

Great! Thanks a lot! (Mmm)
 
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