Anew 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.

Step-by-step explanation:

Data given and notation  

The info given by the problem is:

the random sample taken

represent the population mean

represent the population standard deviation

The critical region on this case is so then if the value of 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:  

Alternative hypothesis:  

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:

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 the significance level.

So we can define like this:

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

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

y = x + 95

y = 42 + 95

y=137 mm hg