A new curing process developed for a certain type of cement...

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 120 kilograms. To test the hypothesis that $\mu=5000$ against the alternative that $\mu<5000$, a random sample of 50 pieces of cement is tested. The critical region is defined to be $\bar{x}<4970$. (a) Find the probability of committing a type I error when $H_0$ is true. (b) Evaluate $\beta$ for the alternatives $\mu=4970$ and $\mu=4960$.

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 120 kilograms. To test the hypothesis that $\mu=5000$ against the alternative that $\mu<5000$, a random sample of 50 pieces of cement is tested. The critical region is defined to be $\bar{x}<4970$.
(a) Find the probability of committing a type I error when $H_0$ is true.
(b) Evaluate $\beta$ for the alternatives $\mu=4970$ and $\mu=4960$.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to assess two competing claims about a population parameter. Typically, a null hypothesis (representing no effect or a status quo) is compared against an alternative hypothesis (indicating the presence of an effect or deviation). Decisions to reject or not reject the null hypothesis are made based on the probability of observing the sample data under the assumption that the null hypothesis is true.

Type I Error

A Type I error occurs when the null hypothesis is incorrectly rejected, meaning that one concludes there is an effect or difference when, in reality, none exists. The probability of a Type I error is denoted by ?, and it defines the level of significance in hypothesis testing. It represents the risk of falsely claiming a change in the underlying parameter.

Type II Error

A Type II error happens when the test fails to reject a false null hypothesis, meaning that one does not detect an effect or difference that truly exists. The probability of a Type II error is denoted by ?. Assessing ? helps researchers understand the test’s power, which is the likelihood of correctly rejecting a false null hypothesis.

Sampling Distribution of the Sample Mean

The sampling distribution of the sample mean is the probability distribution of the sample means computed from multiple independent samples of the same size drawn from the same population. Under the Central Limit Theorem, this distribution tends toward a normal distribution, even if the population itself is not normally distributed, provided that the sample size is sufficiently large.

Standard Error

The standard error is a measure of the variability or dispersion of the sample mean around the true population mean. It is calculated by dividing the population standard deviation by the square root of the sample size. The standard error indicates the precision of the sample mean as an estimate of the population mean.

Critical Region

The critical region is a set of values for the test statistic that leads to the rejection of the null hypothesis. It is defined based on the chosen significance level (?) and represents the outcomes that are considered extreme under the assumption that the null hypothesis is true. Setting the critical region helps determine whether the observed data provides sufficient evidence against the null hypothesis.

Transcript

00:01 In problem number 17, it says a new curing process developed for a certain type of cement results in a mean compressive strength of 5 ,000 kilograms per square centimeter with a standard deviation of 120 kilograms.

00:13 To test the hypothesis that mu equals 5 ,000 against the alternative, mu is less than 5 ,000.

00:18 A random sample of 50 pieces of cement is tested.

00:21 The critical region is defined to be x bar is less than 4970.

00:26 So here i have basically everything written down that we need.

00:29 The population mean is 5 ,000, standard deviation is 120, and the sample size is 50.

00:36 And then the critical region means that's where we're going to reject.

00:39 So if x bar is less than 4970, then we're going to reject.

00:43 So the probability of a type 1 error is that we reject h0, given that h0 is true.

00:50 So we're finding the probability, well, the only way we can reject is if x bar is less than 4970.

00:55 So we're finding the probability that x bar is less than 4970, given that the mean is in fact 5 ,000.

01:04 Okay, so for this, if we draw a little curve, the mu is 5 ,000, and here we have 4970 somewhere over here.

01:14 And we're going to shade, we're going to find this little area here.

01:16 So we do the normal cdf, the normal distribution, so cdf, of negative infinity, all the way up to 4970, and the mean is 5 ,000.

01:27 And the standard deviation, now we're going to use the central limit theorem here.

01:32 It's going to be 120, which is the standard deviation of the population, divided by the square root of your sample size.

01:39 So 120 divided by square root of 50.

01:42 Okay, so from there, you're in your calculator, so we'll go second vars or in any sort of software, or you can convert them to z scores and use a table.

01:50 That's always, you know, possibility.

01:53 A little harder, though.

01:54 So negative e99 will be our lower bound, 49, 70 will be our upper bound.

02:02 The mean is 5 ,000, and the standard deviation is 120 divided by square root of 50.

02:12 So the alpha, the probability of a type 1 error, is about 0 .0385.

02:28 Part b, we're finding the beta, which is the probability of a type 2 error.

02:34 So that means that h not is false and we're going to fail to reject.

02:38 And we're finding beta for the alternatives, mu equals 4970 and mu equals 4960.

02:46 Okay, so 4970, that is where our cutoff point was...