Fry Brothers Heating and Air Conditioning, Inc. employs...

Fry Brothers Heating and Air Conditioning, Inc. employs Larry Clark and George Murnen to make service calls to repair furnaces and air conditioning units in homes. Tom Fry, the owner, would like to know whether there is a difference in the mean number of service calls they make per day. A random sample of 40 days last year showed that Larry Clark made an average of 4.77 calls per day, with a standard deviation of 1.05 calls per day. For a sample of 50 days George Murnen made an average of 5.02 calls per day, with a standard deviation of 1.23 calls per day. At the .05 significance level, is there a difference in the mean number of calls per day between the two employees? What is the $p$ -value?

Key Concepts

p-Value

The p-value is the probability of observing the test statistic, or one more extreme, under the assumption that the null hypothesis is true. A small p-value (typically less than the significance level) provides evidence against the null hypothesis, suggesting that the observed difference is unlikely to have occurred by chance.

Hypothesis Testing and Significance Level

Hypothesis testing is a systematic method for evaluating evidence against a specified null hypothesis, which typically states that there is no effect or difference. The significance level, often set at a predetermined value like 0.05, represents the threshold risk of incorrectly rejecting the null hypothesis, guiding the decision on whether an observed effect is statistically significant.

Standard Error Calculation

The standard error in a two-sample test measures the variability in the difference between sample means. It is computed by taking the square root of the sum of the variances from each sample divided by their respective sample sizes, allowing the difference to be standardized for comparison against a t-distribution.

Two?Sample t-Test

A two?sample t-test is used to compare the means of two independent groups to assess whether any statistically significant difference exists between them. It involves calculating a t-statistic by standardizing the difference between the sample means with respect to the variability and sample sizes of each group.

Degrees of Freedom

Degrees of freedom determine the shape of the t-distribution used in the test and are often approximated via the Welch-Satterthwaite formula when the groups have unequal variances. This approximation adjusts the effective sample size and influences the critical values needed to assess significance.

Transcript

00:01 So we want to use a 5 % significance level to figure out if these two people, larry and george, if they have a difference in the number of calls that are made.

00:13 And so we're going to assume, we'll call him one and call him two.

00:17 We're going to assume that the two are equal, that their number of calls that they make per day is equal, and alternately that they're different.

00:24 So we're not doing a direction test.

00:26 We're doing a two -tail test.

00:28 And we're going to be able to use z values.

00:29 So before i even write down the information, and their sample sizes are sufficiently large enough, so we want to have the 5 % split between the two tails, so 0 .025 and 0 .025 down here to add up 5%.

00:44 And we will be rejecting our null down here and rejecting the null down here.

00:50 And that z value will be 1 .96, and this z value is negative 1 .96.

00:56 So let's write down the data that we have for each of these two.

00:59 And we have for larry that larry has a mean of 4 .77 and has a standard deviation for his days, the number of calls being 1 .05.

01:13 And the sample size we have from larry is 40, 40 days of looking at his number of calls.

01:21 And for george, we have a total of 50 calls, or 50 days that we're looking at.

01:27 And his average over those 50 days was a mean of 5 .02 with a sample standard deviation of 1 .23.

01:37 Now, again, due to the fact that each of these is greater than are equal to 30, your textbook allows us to use a z value to estimate this.

01:47 Some books won't, but your book allows that.

01:49 So let's calculate that z value.

01:52 And so we will take the 4 .77 minus the 5 .0 .3.

01:57 So we can see our test statistic will be negative, and then divided by, and we need to put in our standard error for each of these two individually added together...

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