A test of $H _ { 0 } : \mu = 53$ versus $H _ { 1 } : \mu

Question

A test of $H _ { 0 } : \mu = 53$ versus $H _ { 1 } : \mu < 53$ is performed using a significance level of $\alpha = 0.01$. The $P$-value is $0.046$.
If the true value of $\mu$ is 46 , does the conclusion result in a Type I error, a Type II error, or a correct decision?
A) Correct decision
B) Type I error
C) Type II error

Quizplus · Accepted Answer

Given, 
True value of population mean: $\mu = 46$
Null hypothesis: $H _ { 0 } : \mu = 53$
Alternative hypothesis: $H _ { 1 } : \mu < 53$
Significance level: $\alpha = 0.01$
P-value: 0.046

As the P-value (0.046) is less than the significance level ($\alpha = 0.01$), we reject the null hypothesis. This means that we conclude that the population mean is smaller than 53.

If the true value of $\mu$ is 46, then the true population mean is actually smaller than 53, and our conclusion is correct. Thus, the decision is a correct decision and not a Type I error.

If we had concluded that the population mean is greater than 53, that would have been a Type I error.

If the true value of $\mu$ was not equal to 46 but some value greater than 53, then our conclusion would have been a Type II error, as we would have failed to reject the null hypothesis when it was actually false.

A Test Of H0:μ=53H _ { 0 } : \mu = 53H0​:μ=53

A Test Of $H _ { 0 } : \mu = 53$