Each month the National Association of Purchasing Managers...

Each month the National Association of Purchasing Managers publishes the NAPM index. One of the questions asked on the survey to purchasing agents is: Do you think the economy is expanding? Last month, of the 300 responses 160 answered yes to the question. This month, 170 of the 290 responses indicated they felt the economy was expanding. At the .05 significance level, can we conclude that a larger proportion of the agents believe the economy is expanding this month?

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical procedure used to decide whether there is enough evidence in a sample of data to infer that a certain condition holds true for the entire population. It involves setting up a null hypothesis (a statement of no effect or no difference) and an alternative hypothesis (what you want to test for), and then using test statistics to decide whether to reject the null hypothesis.

Null and Alternative Hypotheses

The null hypothesis is a default statement that there is no effect or no difference between groups, while the alternative hypothesis represents what the researcher seeks to provide evidence for – in this case, that the proportion has increased. Clearly defining these hypotheses is crucial as they determine the direction and type of the statistical test used.

Two-Proportion Z-Test

The two-proportion z-test is a statistical method used to compare the proportions of successes (or a particular outcome) between two independent groups. It involves calculating a z-score that measures the distance between the sample proportions, relative to what would be expected under the null hypothesis, using the standard error of the difference between the proportions.

Standard Error

The standard error measures the variability of the difference between the sample proportions. It is based on the pooled data under the assumption that the null hypothesis is true. A correct computation of the standard error is essential for calculating the z-statistic, which in turn is used to assess the significance of the observed difference.

Significance Level

The significance level, often denoted as alpha, is the threshold for rejecting the null hypothesis. It represents the probability of making a Type I error, which is the error of rejecting a true null hypothesis. A commonly used significance level is 0.05, meaning there is a 5% risk of concluding that a difference exists when there is none.

p-Value

The p-value is the probability, under the null hypothesis, of observing a test statistic as extreme as, or more extreme than, the value computed from the sample data. It provides a measure of the evidence against the null hypothesis; if the p-value is less than the significance level, the null hypothesis is rejected, indicating that the observed effect is statistically significant.

Transcript

00:01 So we want to do a 5 % significance level test, and we will be assuming that the proportion who say that the economy is expanding last month is equal to that for this month.

00:17 So this is last month and this is this month.

00:21 And alternately, that this month's proportion is higher.

00:26 And so our test values, or i should say our p values for last month was to have 160 people out of 300 says yes, said yes, as opposed to this month we had 170 out of 290 said yes, and we will need our pooled p, which is the combination of the two groups.

00:52 So there are 590 people and we would have 330 yes.

00:57 And we want to find out if we have evidence at a 5 % significance level that this second proportion is significantly higher.

01:05 And by the way, this one is approximately 0 .53.

01:09 It's 0 .53 repeating.

01:12 And this one is approximately 0 .586.

01:15 And this one is approximately almost 56 % 0 .559.

01:21 But i'll use these actual numbers.

01:23 And so we want a one -tale test, and this is a z test, a two -proportioned z test, and we want to put all 5 % at one -tail since we're doing a one -tail test, and that z value will correspond to negative 1 .645.

01:40 So we will be rejecting our no if we have a test statistic that is less than this value, and we'll fail to reject if it's on the opposite end.

01:50 And so let's calculate that z value.

01:54 That z value is to take the difference in those two proportions.

02:02 And then i'm just going to put the notation down.

02:05 We're going to take that pooled p and then one minus that pooled p.

02:10 So i'm not going to write this fraction down...

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