The Engineering Department at Sims Software, Inc., recently...

The Engineering Department at Sims Software, Inc., recently developed two chemical solutions designed to increase the usable life of computer disks. A sample of disks treated with the first solution lasted $86,78,66,83,84,81,84,109,65,$ and 102 hours. Those treated with the second solution lasted $91,71,75,76,87,79,73,76,79,78,87,90,76,$ and 72 hours. At the .10 significance level, can we conclude that there is a difference in the length of time the two types of treatment lasted?

Key Concepts

P-Value and Decision Making

The p-value represents the probability of obtaining a test statistic at least as extreme as the one observed, given that the null hypothesis is true. It is compared against the significance level; if the p-value is less than the significance level, the null hypothesis is rejected, indicating significant evidence in favor of a difference between the treatment groups.

Null and Alternative Hypotheses

Formulating the null hypothesis (typically stating that there is no difference between the group means) and the alternative hypothesis (indicating that a difference exists) is a crucial step in the testing process. They provide a decision framework for determining whether the observed data are consistent with the initial assumption of no difference.

Two-Sample t-Test

The two-sample t-test is used to compare the means of two independent groups to determine if there is statistical evidence that the associated population means are significantly different. This test assumes that the samples are drawn from normally distributed populations and considers variations in sample sizes and variance between groups.

Hypothesis Testing

Hypothesis testing is a statistical process used to decide whether a specific claim about a population parameter is supported by sample data. It involves formulating a null hypothesis and an alternative hypothesis, then using test statistics and probability calculations to determine whether to reject the null hypothesis.

Significance Level

The significance level, often denoted by ?, is the threshold probability set by the researcher to determine when to reject the null hypothesis. In this context, a significance level of 0.10 indicates that there is a 10% risk of rejecting the null hypothesis when it is actually true.

Transcript

00:01 So we'll be assuming that the mean of the two treatments has last the same amount of time and alternately that they're different.

00:09 We aren't doing a particular one -sided test.

00:12 So we have a 10 % significance level that we're asked to use and we will have to be using a two sample t test.

00:21 We also are going to be assuming that our two standard deviations are approximately equal for this evaluation.

00:29 And so when i put my data into my calculator, i found that for the first sample.

00:35 We had a sample size on our first group of 10.

00:39 And our second group, we had a sample size of 14.

00:41 So they weren't equal, but that's not required for this technique either.

00:45 And we have our means and our sample standard deviations.

00:48 Let me just get the symbols down here, and then we can pick off what those numbers are.

00:52 So the first one has a mean of 83 .8.

00:56 And the second one had a mean of, 79 .28, call it 6.

01:02 And the sample standard deviation for the first one is 13 .67.

01:09 And our second one has a standard deviation of 6 .707.

01:15 So this technique, we're assuming these two are equal.

01:19 And there are other techniques that you can use with t test that don't assume they're equal, but that's the technique that your book has shown.

01:26 So we are required.

01:28 At this point to find that p that pooled standard deviation and or the pooled variance in this case and we know we do that by taking one less than the sample size of the first and then times the variance of the first so we're finding kind of a weighted a pooled variance and then one less than this sample size times this standard deviation squared or its variance and then we divide that by the sum of the two sample sizes less two.

02:02 And this ends up being our degrees of freedom as well, which is a degrees of freedom of 22.

02:07 And while we're at it, why don't i just get my critical values for this setting since i just put the degrees of freedom.

02:16 So we know since it's a 10 % significance level, we will have half of that at each tail with our two tail tests and have 0 .05 here, 0 .05 there.

02:25 And let's find out what those test statistics are that will cause us to reject our null.

02:32 And so with 22 degrees of freedom and 5 % in the upper tail, we have this value is 1 .717.

02:40 And this one will be negative 1 .717.

02:43 And we would be rejecting the null here and rejecting the null if the value is up there as well...

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