Registered nurses earned an average annual salary of 69,110....

Registered nurses earned an average annual salary of $69,110. For that same year, a survey was conducted of 41 California registered nurses to determine if the annual salary is higher than $69,110 for California nurses. The sample average was $71,121 with a sample standard deviation of $7,489. Conduct a hypothesis test.

Key Concepts

p-Value

The p-value is the probability of obtaining test results at least as extreme as the ones observed, assuming that the null hypothesis is true. It serves as a measure of the evidence against the null hypothesis. A small p-value indicates strong evidence against H?, suggesting that the observed sample provides sufficient reason to reject the null hypothesis in favor of the alternative hypothesis.

Degrees of Freedom

Degrees of freedom refer to the number of independent values in a statistical calculation that are free to vary when estimating a parameter. In the context of a one-sample t-test, degrees of freedom are typically calculated as the sample size minus one. They are crucial for determining the appropriate t-distribution to use when obtaining critical values and p-values.

Significance Level

The significance level, often denoted by alpha (?), is the threshold used to determine whether the p-value is low enough to reject the null hypothesis. Common significance levels include 0.05, 0.01, and 0.10. It represents the probability of making a Type I error, which is rejecting the null hypothesis when it is actually true.

One-Sample t-Test

A one-sample t-test is used to evaluate whether the mean of a sample is statistically different from a known or hypothesized population mean when the population standard deviation is unknown. This test applies the t-distribution to approximate the sampling distribution of the sample mean, particularly when the sample size is small or the population variance is not known.

Null and Alternative Hypotheses

This concept involves setting up two competing statements about a population parameter. The null hypothesis (H?) typically represents the status quo or a claim of no effect, while the alternative hypothesis (H? or Ha) reflects the claim to be tested. In hypothesis testing for a mean, H? might state that the population mean is a certain value, whereas H? proposes that it is different (or greater, or less, depending on the direction of the test).

Test Statistic

The test statistic is a numerical summary that represents the degree of agreement between the sample data and the null hypothesis. In a one-sample t-test, it is calculated by taking the difference between the sample mean and the hypothesized population mean, divided by the standard error of the mean. This value is then compared against critical values from the t-distribution to decide whether to reject the null hypothesis.

Transcript

00:01 Hello, welcome to the video, and today we're going to be talking about hypothesis tests, and specifically how to know when our evidence does or does not sufficiently support our claim.

00:11 So in this problem, we're told that the average annual salary of nurses is $69 ,110, so we can translate that into a null hypothesis by saying that mew, or the average annual salary of nurses, is equal to 69 ,110.

00:36 And it is proposed that the true average is greater than this value here.

00:44 So we can translate that into an alternative hypothesis by saying that mu is greater than 69 ,110.

00:54 So if we want to visualize this here, let's say we have a distribution with a peak value, at the proposed 69 ,110.

01:08 With our hypothesis test here, we hope to find evidence to show that the true value of mu is actually somewhere over here and not close to what is proposed here.

01:22 So in other words, we'll be doing a right -tailed hypothesis test here.

01:28 And we will be doing this hypothesis test at a five alpha level.

01:35 So to test these hypotheses we do a survey of 41 nurses and so just, yeah, we're told that a sample size of 41.

01:51 We're told that the sample mean was 71 ,121 with a sample standard deviation of 7 ,1 ,121 with a sample standard deviation of 7 ,000.

02:10 So this is the information we gathered from our sample.

02:15 And with this information here and the information we're given here, we can go to our calculators to find a test statistic and p -value to determine what we should do with our hypotheses...

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