Test the given claim. Identify the null hypothesis,...

Test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, P-value or critical value(s), then state the conclusion about the null hypothesis, as well as the final conclusion that addresses the original claim. A trial was conducted with 75 women in China given a 100-yuan bill, while another 75 women in China were given 100 yuan in the form of smaller bills (a 50-yuan bill plus two 20-yuan bills plus two 5-yuan bills). Among those given the single bill, 60 spent some or all of the money. Among those given the smaller bills, 68 spent some or all of the money (based on data from "The Denomination Effect," by Raghubir and Srivastava, Journal of Consumer Research, Vol. 36). We want to use a 0.05 significance level to test the claim that when given a single large bill, a smaller proportion of women in China spend some or all of the money when compared to the proportion of women in China given the same amount in smaller bills. a. Test the claim using a hypothesis test. b. Test the claim by constructing an appropriate confidence interval. c. If the significance level is changed to 0.01, does the conclusion change?

Test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, P-value or critical value(s), then state the conclusion about the null hypothesis, as well as the final conclusion that addresses the original claim.
A trial was conducted with 75 women in China given a 100-yuan bill, while another 75 women in China were given 100 yuan in the form of smaller bills (a 50-yuan bill plus two 20-yuan bills plus two 5-yuan bills). Among those given the single bill, 60 spent some or all of the money. Among those given the smaller bills, 68 spent some or all of the money (based on data from "The Denomination Effect," by Raghubir and Srivastava, Journal of Consumer Research, Vol. 36). We want to use a 0.05 significance level to test the claim that when given a single large bill, a smaller proportion of women in China spend some or all of the money when compared to the proportion of women in China given the same amount in smaller bills.
a. Test the claim using a hypothesis test.
b. Test the claim by constructing an appropriate confidence interval.
c. If the significance level is changed to 0.01, does the conclusion change?

Key Concepts

Significance Level

The significance level, often denoted by alpha (?), is the threshold for making a decision in hypothesis testing. It represents the probability of rejecting the null hypothesis when it is actually true (Type I error). Common values are 0.05 and 0.01, which indicate the risk levels of making a false positive error, with lower values implying a more stringent test.

Drawing Conclusions in Hypothesis Testing

Drawing conclusions involves interpreting the test statistic, P-value, confidence interval, and significance level to decide whether to reject or fail to reject the null hypothesis. This process also includes considering practical significance and the context of the original claim to provide a final statement about whether the observed data support or refute the initial hypothesis.

Confidence Interval

A confidence interval provides a range of plausible values for the parameter of interest based on the sample data. When comparing two proportions, it estimates the range within which the true difference between the population proportions is likely to fall. If the interval does not include the hypothesized difference (often zero), it provides evidence against the null hypothesis.

Critical Value

A critical value is a cutoff point on the test statistic distribution which corresponds to the chosen significance level. For hypothesis tests, if the calculated test statistic falls beyond this critical value, the null hypothesis can be rejected. Critical values provide an alternative way to assess the statistical significance of the results.

P-Value

The P-value represents the probability of obtaining test results at least as extreme as the results actually observed, assuming that the null hypothesis is correct. It is used as a benchmark to decide whether the observed data are sufficiently unlikely under the null hypothesis. If the P-value is lower than the predetermined significance level, the null hypothesis is rejected.

Test Statistic

The test statistic is a standardized value calculated from sample data under the assumption that the null hypothesis is true. In the context of comparing two proportions, it is computed using the difference of the sample proportions, the pooled proportion, and the standard error. It quantifies how many standard errors the observed difference is away from the hypothesized difference of zero.

Two-Proportion Z-Test

This statistical test compares the difference between two sample proportions to see if it is statistically significant. It involves calculating a test statistic which measures the difference in proportions relative to the variability expected if the null hypothesis were true. The two-proportion z-test is used when independent samples are considered and can accommodate large-sample approximations.

Alternative Hypothesis

The alternative hypothesis is what the researcher aims to support and reflects the presence of an effect or a difference. When comparing proportions, it posits that the proportions are not equal or, in a one-sided test, that one group’s proportion is either higher or lower than the other’s. It directly contrasts the null and is used to assess if observed data indicate a significant deviation from the status quo.

Null Hypothesis

In hypothesis testing, the null hypothesis represents a default assumption where any observed difference is attributed to chance. In the context of comparing two proportions, it generally states that there is no effect or no difference between the two groups. This provides a baseline which the evidence must overcome to support an alternative claim.

Transcript

00:01 In this exercise we're going to be considering a trial that was conducted with 75 women in china who were given a 100 yuan bill, while another 75 women in china were given 100 yuan in the form of smaller bills.

00:24 So we're told to test the claim that when a single single one, large bill is given a smaller proportion of women in china spend some or all the money compared to the proportion of the money, the proportion of women in china given the same amount in smaller bits.

00:56 If you're given smaller bits, you tend to spend more.

01:01 Or in other words, when you're given a single large bill, you tend to spend less.

01:07 So let's see.

01:09 How are we going to do that? and we're going to do it in two ways using the hypothesis test and using the confidence interval method.

01:17 So when we consider the data, we have two samples.

01:25 The first sample was given a single bill, large bill.

01:39 And the second sample was given the same amount of money but in smaller bills.

01:47 So out of the 75 women that were given a large bill, 60 of them spent some or all of the money.

02:06 And out of the 75 women that were given smaller bills, 68 spent some or all of the money.

02:19 So at the 0 .01, 0 .05 level of significance, we're going to test the claim that a smaller portion, a smaller proportion of women who are given the large bill spend some or all of the money.

02:50 So the null hypothesis will be p1 is equal to p2, and the alternative hypothesis is p1 is less than p2...

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