Question

A new curing process developed for a certain type of cement results in a mean compressive strength of 5000 kilograms per square centimeter with a standard deviation of 100 kilograms. To test the hypothesis that µ = 5000 against the alternative that µ < 5000, a random sample of 25 pieces of cement is tested. The critical region is defined to be x<4970. Find the probability of committing a type I error when H0 is true.

Answer 1

Answer:

$\alpha =0.0668$

Step-by-step explanation:

Data given and notation  

The info given by the problem is:

$n=25$ the random sample taken

$\mu =5000$ represent the population mean

$\sigma =100$ represent the population standard deviation

The critical region on this case is $\bar X$ so then if the value of $\bar X \geq 4970$ we fail to reject the null hypothesis. In other case we reject the null hypothesis

Null and alternative hypotheses to be tested  

We need to conduct a hypothesis in order to determine if the true mean is 5000, the system of hypothesis would be:  

Null hypothesis:$\mu = 5000$  

Alternative hypothesis:$\mu \neq 5000$  

Let's define the random variable X ="The compressive strength".

We know from the Central Limit Theorem that the distribution for the sample mean is given by:

$\bar X \sim N(\mu , \frac{\sigma}{\sqrt{n}})$

Find the probability of committing a type I error when H0 is true.

The definition for type of error I is reject the null hypothesis when actually is true, and is defined as $\alpha$ the significance level.

So we can define $\alpha$ like this:

$\alpha= P(Error I)= P(\bar X$

And in order to find this probability we can use the Z score given by this formula:

$Z=\frac{\bar X -\mu}{\frac{\sigma}{\sqrt{n}}}$

And the value for the probability of error I is givn by:

$\alpha= P(\bar X$