10. Hypothesis Testing for Two Samples

Two Proportions

10. Hypothesis Testing for Two Samples

Two Proportions

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Multiple Choice

The data below is taken from two random, independent samples. Calculate the margin of error for a 99% confidence interval for the difference in population proportions.
$x sub 1 equals 87$ , $x sub 2 equals 68$
$n sub 1 equals 120$ , $n sub 2 equals 115$

0.061

0.062

0.158

0.060

Verified step by step guidance

Step 1: Understand the problem. We are tasked with calculating the margin of error for a 99% confidence interval for the difference in population proportions. The given data includes the sample successes (x₁ = 87, x₂ = 68) and sample sizes (n₁ = 120, n₂ = 115).

Step 2: Calculate the sample proportions for each group. The sample proportion for group 1 is p̂₁ = x₁ / n₁, and for group 2, it is p̂₂ = x₂ / n₂. Use these formulas to compute the sample proportions.

Step 3: Compute the standard error (SE) for the difference in proportions. The formula for SE is: SE = sqrt((p̂₁(1 - p̂₁) / n₁) + (p̂₂(1 - p̂₂) / n₂)). Substitute the values of p̂₁, p̂₂, n₁, and n₂ into this formula.

Step 4: Determine the critical value (z*) for a 99% confidence level. For a 99% confidence interval, the z* value corresponds to the 99% level of confidence, which is approximately 2.576. This value is derived from the standard normal distribution.

Step 5: Calculate the margin of error (ME). The formula for ME is: ME = z* × SE. Multiply the critical value (z*) by the standard error (SE) to find the margin of error.

9:27m

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Textbook Question

In Exercises 5–8, use (a) randomization and (b) bootstrapping for the indicated exercise from Section 9-1. Compare the results to those obtained in the original exercise.

Exercise 9 in Section 9-1 “Cell Phones and Handedness”

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Multiple Choice

A human resources department is comparing two employee training programs to see if they lead to different pass rates on a required certification exam. They randomly select two groups of employees. In Program A, 16 out of 20 employees passed the exam. In Program B, 30 out of 40 employees passed. Are the basic conditions met to conduct a 2-proportion hypothesis test?

A school administrator wants to compare the proportion of students who passed a standardized math exam in two different schools by taking samples from 2 classes. Assume the samples are random and independent. Calculate the $z$ -score for testing whether there is a significant difference in the population proportions of student passing rates, but do not find a $P$ -value or draw a conclusion for the hypothesis test.

Class A: 72 out of 120 students passed.

Class B: 65 out of 100 students passed.

A researcher using a survey constructs a 90% confidence interval for a difference in two proportions. According to the data, they calculate ${\hat{p}}_{1} - {\hat{p}}_{2} = 0.09$ with a margin of error of 0.07. Should they reject or fail to reject the claim that there is no difference in these two proportions?

Denomination Effect 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.

108

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Textbook Question

c. If the significance level is changed to 0.01, does the conclusion change?

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Textbook Question

Clinical Trials of OxyContin OxyContin (oxycodone) is a drug used to treat pain, but it is well known for its addictiveness and danger. In a clinical trial, among subjects treated with OxyContin, 52 developed nausea and 175 did not develop nausea. Among other subjects given placebos, 5 developed nausea and 40 did not develop nausea (based on data from Purdue Pharma L.P.). Use a 0.05 significance level to test for a difference between the rates of nausea for those treated with OxyContin and those given a placebo.

c. Does nausea appear to be an adverse reaction resulting from OxyContin?

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Textbook Question

b. Test the claim by constructing an appropriate confidence interval.

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Struggling with Statistics?

The data below is taken from two random, independent samples. Calculate the margin of error for a 99% confidence interval for the difference in population proportions.x1=87x_1=87, x2=68x_2=68n1=120n_1=120, n2=115n_2=115

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Two Proportions practice set

The data below is taken from two random, independent samples. Calculate the margin of error for a 99% confidence interval for the difference in population proportions.
$x sub 1 equals 87$ , $x sub 2 equals 68$
$n sub 1 equals 120$ , $n sub 2 equals 115$