The Tampa Bay (Florida) Area Chamber of Commerce wanted to...

Name: Problem 18, Chapter 11: Basic Statistics for Business and Economics
Uploaded: 2022-03-31T20:29:08-08:00
Duration: 6 min 3 s
Channel: Sheryl Ezze
Description: The Tampa Bay (Florida) Area Chamber of Commerce wanted to know whether the mean weekly salary of nurses was larger than that of school teachers. To investigate, they collected the following information on the amounts earned last week by a sample of school teachers and nurses.

The Tampa Bay (Florida) Area Chamber of Commerce wanted to know whether the mean weekly salary of nurses was larger than that of school teachers. To investigate, they collected the following information on the amounts earned last week by a sample of school teachers and nurses.

Key Concepts

Type I and Type II Errors

When conducting hypothesis tests, there is always a risk of making errors. A Type I error occurs when the null hypothesis is incorrectly rejected when it is actually true, while a Type II error happens when the test fails to reject a false null hypothesis. Balancing these errors is essential for designing effective tests and accurately interpreting results.

Significance Level and p-Value

The significance level is a threshold set by the researcher, commonly 0.05, that determines the probability at which the null hypothesis will be rejected. The p-value is the probability of obtaining a result at least as extreme as the observed one, assuming the null hypothesis is true. A small p-value indicates that the observed difference is unlikely to have occurred by chance, supporting the alternative hypothesis.

Sampling Distribution

The concept of a sampling distribution refers to the probability distribution of a statistic, like the mean, obtained from random samples drawn from a population. Understanding the sampling distribution is critical in hypothesis testing because it provides the basis for determining the probability of observing a sample statistic, which allows for assessments of significance.

Two-Sample t-Test

The two-sample t-test is used for comparing the means of two independent groups to assess whether their population means are significantly different. This test takes into account the variability within each sample and the size of each group so that one can determine if the observed difference in sample means is unlikely to occur by chance.

Hypothesis Testing

Hypothesis testing is a statistical method that allows researchers to use sample data to evaluate a claim or assumption about a population parameter. In the context of comparing means, it involves setting up a null hypothesis (usually stating no difference between the means) and an alternative hypothesis (stating a specific difference or effect) and then determining whether the observed data provide strong enough evidence to reject the null hypothesis.

Transcript

00:01 So we're looking at data from the tampa area, and we want to look at the teacher salary, mean teacher salary, compared to nurses ' salaries.

00:12 And we're going to assume that they're equal, and alternately that the nurses is higher than the teachers.

00:19 And we are doing this lesson or this problem, and it does not give us a significance level.

00:26 So we'll go through and find the p value to make our decision.

00:30 We need to find first of all we know that our x bar for the teachers i found that to be $826 .75 and the mean i found for the nurses is $835 and 75 and the first for the teachers standard deviation that was $22 .84 and for the nursing salaries.

01:03 Those were $34 in, well put, whoops, 34 and 40 .4.

01:11 And this came out to be, would round off to three decimal places to that.

01:16 And the sample size for our first group was 12 for the teachers.

01:22 And for the nursing group, we only have eight.

01:26 Now, we're also going to assume that these two standard deviations are approximately equal.

01:32 So we're going to need to find that pooled variance.

01:36 And so let's find that now.

01:40 And we're going to take one less than our sample size, 11 times that standard deviation of 22 .840 squared plus one less than this sample size.

01:51 So seven times this variance.

01:54 So 34 .404.

01:57 And we'll square that.

01:58 And then we're going to divide that by the sum of the two sample sizes, less two.

02:05 And this is kind of a dual meaning because that denominator is also our degrees of freedom.

02:12 And we have 18 degrees of freedom.

02:14 And when we assume that these two standard deviations are equal.

02:18 So let's calculate that pooled value.

02:21 So i have left parentheses 11 times that 22 .84 squared plus.

02:28 7 times that 34 .404 squared, close that parentheses, and then divided by 18.

02:37 And that pulled value for the variance is 779 .09 .98...

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