You have been asked to determine if two different production...

You have been asked to determine if two different production processes have different mean numbers of units produced per hour. Process 1 has a mean defined as $\mu_1$ and process 2 has a mean defined as $\mu_2$. The null and alternative hypotheses are as follows: $$ \begin{aligned} & H_0: \mu_1-\mu_2=0 \\ & H_1: \mu_1-\mu_2>0 \end{aligned} $$ Use a random sample of 25 observations from process 1 and 28 observations from process 2 and the known variance for process 1 equal to 900 and the known variance for process 2 equal to 1,600 . Can you reject the null hypothesis using a probability of Type I error $\alpha=0.05$ in each case? a. The process means are 50 and 60 . b. The difference in process means is 20 . c. The process means are 45 and 50. d. The difference in process means is 15 .

You have been asked to determine if two different production processes have different mean numbers of units produced per hour. Process 1 has a mean defined as $\mu_1$ and process 2 has a mean defined as $\mu_2$. The null and alternative hypotheses are as follows:
$$
\begin{aligned}
& H_0: \mu_1-\mu_2=0 \\
& H_1: \mu_1-\mu_2>0
\end{aligned}
$$
Use a random sample of 25 observations from process 1 and 28 observations from process 2 and the known variance for process 1 equal to 900 and the known variance for process 2 equal to 1,600 . Can you reject the null hypothesis using a probability of Type I error $\alpha=0.05$ in each case?
a. The process means are 50 and 60 .
b. The difference in process means is 20 .
c. The process means are 45 and 50.
d. The difference in process means is 15 .

Key Concepts

Significance Level

The significance level, often denoted as alpha (?), is the probability threshold set by the researcher for rejecting the null hypothesis. It defines the acceptable risk of making a Type I error, and common choices include 0.05 or 0.01, reflecting a 5% or 1% risk, respectively.

Sampling Distribution of the Difference Between Means

When comparing two independent samples, the sampling distribution of the difference between the sample means reflects how that difference would vary from sample to sample. This distribution is used to standardize the difference, given the known variances and sample sizes, to compute a test statistic for hypothesis testing.

Type I Error

A Type I error occurs when the null hypothesis is wrongly rejected even though it is true. The probability of committing a Type I error is denoted by the significance level alpha (?), and it represents the threshold at which the evidence against the null hypothesis is considered strong enough to reject it.

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether there is enough evidence in a sample of data to infer that a certain condition is true for the entire population. It involves formulating a null hypothesis and an alternative hypothesis, selecting a test statistic, and using a significance level to determine whether to reject the null hypothesis.

Null Hypothesis and Alternative Hypothesis

The null hypothesis (H0) typically represents a position of no effect or no difference, while the alternative hypothesis (H1) represents the presence of an effect or a difference. In tests comparing two means, the null hypothesis generally claims that the difference between the means is zero, whereas the alternative might state that one mean is greater than the other.

Two-Sample Z-Test

A two-sample z-test is used to compare the means of two independent groups when the variances are known and the sample sizes are specified. It involves computing the difference between the two sample means and standardizing this difference to see if it is statistically significant relative to the known variances.

Transcript

00:01 I'll share how to do the first one, and then we won't do the rest of them.

00:03 We'll leave those for you.

00:04 We actually failed to reject the null on all these.

00:07 But here's our original assumption, and alternately we have this mean being higher.

00:16 And we would find our pooled p to begin with, our pooled variance, i should say, our pooled standard deviation.

00:23 And let me quick, i've done a whole bunch of calculations here, so let me quick get what those values are going to be.

00:29 We would end up having our first standard deviation.

00:32 We know our sample size is 25, so we'll go 24, times the first standard deviation, which is 30, and then add to that 25, and the second standard deviation is 28.

00:46 So we need that variance, and then we divide that by the 25 plus 25 minus 2, or that's how many degrees of freedom we have.

00:58 And then we calculate our test statistic.

01:00 Our test statistic will be the difference in the means, divided by and then we have the square root of and we're going to take what this pooled variance is and divided by the first sample size and the second variance which again is the same pooled one because we're assuming they're equal and this test statistic comes out to be comes out to be 0 .73 and so are we have 48 degrees of freedom we want to find the probability that our test statistic with 48 degrees of freedom is greater than or equal to 0 .73.

01:37 And that p value comes out to be 0 .468...

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