According to the N . Y. Times Almanac the mean family size...

Key Concepts

p-value

The p-value is a measure that quantifies the strength of evidence against the null hypothesis. It represents the probability of obtaining a test statistic at least as extreme as the one observed, assuming that the null hypothesis is true. A small p-value (typically below the significance level) indicates that the observed outcome is unlikely under the null hypothesis, thus suggesting sufficient evidence to reject it.

One-Sample t-Test

The one-sample t-test is used to compare the sample mean to a known or hypothesized population mean when the population standard deviation is unknown and the sample size is relatively small. This test calculates a t-statistic by evaluating the difference between the sample mean and the population mean, divided by the standard error of the mean. The t-statistic is then compared to a critical value from the t-distribution to determine whether the observed difference is statistically significant.

Data Validity and Generalizability

Assessing data validity involves determining whether a specific sample result, such as the family sizes in a college math class, accurately reflects a population parameter like the national family size average. While a statistically significant result in a local sample might indicate a difference, it is important to consider factors such as sample representativeness and potential biases before generalizing findings to larger populations. This caution ensures that conclusions drawn from the sample maintain relevance and validity in the broader context.

Null and Alternative Hypotheses

In the context of comparing a sample mean to a known population mean, the null hypothesis typically states that the population mean is equal to the claimed value, while the alternative hypothesis states that the population mean is different (or specifically greater, in a one-tailed test) than that value. This framework helps in systematically assessing whether any observed difference can be attributed to random variability or suggests a true deviation from the claimed population parameter.

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences about population parameters based on sample data. It involves setting up a null hypothesis, which represents the status quo or a specific claim about a population parameter, and an alternative hypothesis, which represents a competing claim. The goal is to determine whether there is sufficient evidence in the sample data to reject the null hypothesis in favor of the alternative, given a predetermined significance level.

Significance Level (?)

The significance level, denoted by ?, represents the probability of rejecting the null hypothesis when it is actually true (a Type I error). It is a threshold set by the researcher before conducting the test, commonly 0.05 or 5%, which helps determine the strength of evidence required for making a statistical decision. A result with a p-value less than or equal to ? leads to the rejection of the null hypothesis, suggesting that the observed effect is statistically significant.

Transcript

00:01 The following is a solution to number 11, and this looks at family size, ideal family size, or ideal number of kids, and we compare it from 1985 to present day.

00:11 Now, this book is a little dated, i think it's the early 2000s, but about a 15 to 20 year difference there.

00:16 And we're given that the sample size was 106.

00:19 The population standard deviation was 1 .2, and the sample for the current sample is the ideal number of children is 2 .6.

00:30 And we're testing at the alpha value of .10 that the ideal number of children hasn't changed from 1985.

00:39 And in 1985, they said 2 .5 children is the ideal number of children.

00:44 So first off, we need to say what we're doing here.

00:47 We're going to do the z test.

00:48 And the reason why we're doing the z test is we're testing for the population mean.

00:54 So it's either going to be the z or the t.

00:56 And we're able to use the z test because we know that population standard.

01:00 Of 1 .2.

01:01 So since we're given sigma, we can go ahead and use the z test.

01:05 So we're going to do a five -step hypothesis test and the first test or the first step is to state our hypotheses and the null hypothesis h -0 is that the population mean is whatever it said it was and then back in itme5 it was 2 .5.

01:22 The alternative is what we're testing.

01:24 So we're testing to see if it's changed.

01:26 Now we don't know if it's going up or down.

01:29 We just are wondering if it's changed.

01:30 So we're going to say not equal to 2 .5.

01:34 Okay, so then steps two and three, this is where we get our z value and our p value.

01:40 And you can get these with formulas and charts and whatnot, but it's much easier if you just use technology.

01:44 So i'm going to use a ti -84 because it works pretty nicely.

01:47 So on a t -i -84, you can use any sort of calculator you want.

01:50 If you go to stat, arrow over to test, it's this first option here, the z -test.

01:56 And summary stats were given.

01:58 So the null hypothesis, we remember, was 2 .5.