Data released by the National Center for Health Statistics...

Data released by the National Center for Health Statistics showed that the mean age at which women had their first child was 25.0 in 2006 (The Wall Street Joumal, February 4 , 2009 ). The reporter, Sue Shellenbarger, noted that this was the first decrease in the average age at which women had their first child in several years. A recent sample of 42 women provided the data in the website file named FirstBirth concerning the age at which these women had their first child. Do the data indicate a change from 2006 in the mean age at which women had their first child? Use $a=.05$.

Key Concepts

Statistical Hypothesis Testing

This is the process of using sample data to evaluate a claim about a population parameter. It involves setting up a null hypothesis, which reflects a status quo condition, and an alternative hypothesis, which represents a deviation from that status quo, then using statistical methods to determine whether the observed data are unlikely under the null hypothesis.

Null and Alternative Hypotheses

The null hypothesis typically states that there is no difference or change from a known value, while the alternative hypothesis posits that there is a difference. In a testing procedure, these hypotheses are mutually exclusive, and evidence from the sample is used to decide whether there is sufficient reason to reject the null hypothesis in favor of the alternative.

Significance Level (?)

The significance level is a predetermined threshold that defines the probability of making a Type I error, which is rejecting the null hypothesis when it is actually true. A common choice is 0.05, meaning there is a 5% risk of incorrectly rejecting the null hypothesis, and it serves as a benchmark for determining statistical significance.

One-Sample Mean Test

A one-sample mean test involves comparing the mean from a sample to a known or hypothesized population mean. This test calculates a test statistic, such as using a t-test when the population standard deviation is unknown or a z-test when it is known, to assess whether the observed sample mean significantly deviates from the hypothesized value.

Transcript

00:01 So in this problem, we are interested in finding if the data indicates a change from 2006 in the mean age at which women had their first child.

00:10 So we are interested in finding a change if there's a change in the mean.

00:14 So our alternative hypothesis is going to be that mu is not equal to the original mean, which is 25.

00:24 And that means our null hypothesis is going to be that mu is equal to 25.

00:31 So with this, we are going to compute a, we're going to find a p value in order to comply to our alpha of 0 .05.

00:42 Also given in the problem, we see that our n is 42.

00:45 And in order to find our x bar and our sample standard deviation, we have to go to excel.

00:55 So here are our descriptive stats.

01:00 There's x bar our standard deviation s and our sample size and here are the formulas that you can use to get those values so using these values we're going to compute a t test statistic and we're using a t test statistic because our standard deviation is a sample standard deviation we do not have a population standard deviation so we are going to take x bar minus the population mean divided by the sample mean over the square root of the sample size, which is equal to, and i'm going to round our x bar to 24 .04, 24 .048, sorry, 24 .048 minus 25 over 5 .885 divided by the square root of 42, and i get a t test statistic of negative 1 .05.

01:59 So what exactly is? does this value mean? if we draw a t distribution, t equals zero lies in the middle, we get that our, we have a t equals negative 1 .05 lies here.

02:23 We're interested in finding the area to the left of this t value.

02:29 So this area here represents the probability.

02:33 The t is less than negative 1 .05, less than are equal to negative 1 .05.

02:39 And how do we compute that? first, we have to find a degrees of freedom.

02:42 We have our n is 42, so our degrees of freedom is equal to n minus 1, which is equal to 41...

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