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Chapter 10: Problem 1

Use the following information to answer the next 15 exercises: Indicate if the hypothesis test is for a. independent group means, population standard deviations, and/or variances known b. independent group means, population standard deviations, and/or variances unknown c. matched or paired samples d. single mean e. two proportions f. single proportion It is believed that 70% of males pass their drivers test in the first attempt, while 65% of females pass the test in the first attempt. Of interest is whether the proportions are in fact equal.

Short Answer

Expert verified
The hypothesis test is for two proportions.

Step by step solution

01

Identify the Type of Data

The data involves proportions of two groups: males and females. Specifically, we have the proportion of males who pass (70%) and the proportion of females who pass (65%).
02

Determine the Parameter to Test

Since the problem involves comparing proportions from two independent groups (males and females), we focus on testing the equality of these two proportions.
03

Choose the Appropriate Hypothesis Test

The question aims to test whether the proportions of males and females passing the driver's test are equal. This corresponds to testing the equality of two proportions.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

two proportions
When we talk about 'two proportions' in statistics, we refer to comparing the proportions of a specific characteristic in two different groups. In the given exercise, these two groups are males and females.

To break it down further:
  • The proportion for males is given as 70%, which means that out of every 100 males, 70 are expected to pass the driver's test on their first attempt.
  • The proportion for females is 65%, indicating that 65 out of 100 females may pass successfully on their first try.
Understanding these proportions is crucial as it forms the basis of our hypothesis test. We're looking to see if these percentages (proportions) are indeed different, or if any observed discrepancy is due to random chance.
independent groups
In hypothesis testing, the concept of 'independent groups' is key to ensuring the validity of our analysis. Groups are independent if the outcome or characteristic being measured in one group does not affect the outcome in the other group.

For our exercise, the two groups are males and females taking a driver's test. These groups are considered independent because:
  • The test results of males do not influence the outcomes for females and vice versa.
  • Both groups are separate with no overlap in membership or shared characteristics that could affect the proportion of first-time passes.
Independence allows for a straightforward comparison of the two proportions, ensuring that any statistical methods applied are based on clean, unbiased data.
proportion comparison
Comparing proportions is the central theme in this hypothesis test. We seek to determine if the proportion of passing males is significantly different from the proportion of passing females.

This type of comparison involves:
  • Setting up a null hypothesis, which generally states that there is no difference between the two proportions. In this case, it assumes that the proportion of males passing is equal to the proportion of females passing.
  • Formulating an alternative hypothesis if we find that these proportions are not equal.
Proportion comparison drives the direction of our statistical test, guiding whether we accept or reject the null hypothesis based on the data provided from independent samples.
statistical test
A 'statistical test' helps in deciding whether the observed data diverge significantly from what was expected under the null hypothesis. For comparing two proportions, as in this exercise, we typically use a test for equality of proportions.

Here's a closer look at what that entails:
  • We calculate a test statistic, which quantifies the difference between the observed proportions relative to what we would expect under the null hypothesis.
  • The decision to reject or not reject the null hypothesis is based on the p-value, which tells us the probability of observing a more extreme difference in proportions under the assumption that the null hypothesis is true.
  • A significant p-value (usually less than 0.05) would lead us to reject the null hypothesis - suggesting a real difference in proportions between males and females passing their driver's tests.
Statistical tests provide a structured methodology to either support or refute theories about population parameters, empowering data-driven decision-making.

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Most popular questions from this chapter

Use the following information to answer the next 12 exercises: The U.S. Center for Disease Control reports that the mean life expectancy was 47.6 years for whites born in 1900 and 33.0 years for nonwhites. Suppose that you randomly survey death records for people born in 1900 in a certain county. Of the 124 whites, the mean life span was 45.3 years with a standard deviation of 12.7 years. Of the 82 nonwhites, the mean life span was 34.1 years with a standard deviation of 15.6 years. Conduct a hypothesis test to see if the mean life spans in the county were the same for whites and nonwhites. At a pre-conceived \(\alpha=0.05,\) what is your: a. Decision: b. Reason for the decision: c. Conclusion (write out in a complete sentence):

Use the following information to answer the next ten exercises. indicate which of the following choices best identifies the hypothesis test. a. independent group means, population standard deviations and/or variances known b. independent group means, population standard deviations and/or variances unknown c. matched or paired samples d. single mean e. two proportions f. single proportion A football league reported that the mean number of touchdowns per game was five. A study is done to determine if the mean number of touchdowns has decreased.

A study is done to determine if students in the California state university system take longer to graduate, on average, than students enrolled in private universities. One hundred students from both the California state university system and private universities are surveyed. Suppose that from years of research, it is known that the population standard deviations are 1.5811 years and 1 year, respectively. The following data are collected. The California state university system students took on average 4.5 years with a standard deviation of 0.8. The private university students took on average 4.1 years with a standard deviation of 0.3.

Use the following information to answer the next two exercises. An experiment is conducted to show that blood pressure can be consciously reduced in people trained in a 鈥渂iofeedback exercise program.鈥� Six subjects were randomly selected and blood pressure measurements were recorded before and after the training. The difference between blood pressures was calculated (after - before) producing the following results: \(\overline{x}_{d}=-10.2 \mathrm{sd}=8.4\) Using the data, test the hypothesis that the blood pressure has decreased after the training. If \(\alpha=0.05,\) the \(p\) -value and the conclusion are a. 0.0014; There is sufficient evidence to conclude that the blood pressure decreased after the training. b. 0.0014; There is sufficient evidence to conclude that the blood pressure increased after the training. c. 0.0155; There is sufficient evidence to conclude that the blood pressure decreased after the training. d. 0.0155; There is sufficient evidence to conclude that the blood pressure increased after the training.

Use the following information to answer the next twelve exercises. In the recent Census, three percent of the U.S. population reported being of two or more races. However, the percent varies tremendously from state to state. Suppose that two random surveys are conducted. In the first random survey, out of 1,000 North Dakotans, only nine people reported being of two or more races. In the second random survey, out of 500 Nevadans, 17 people reported being of two or more races. Conduct a hypothesis test to determine if the population percents are the same for the two states or if the percent for Nevada is statistically higher than for North Dakota. Which distribution (normal or Student's t) would you use for this hypothesis test?
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