/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 65 The average waiting time in a do... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 65

The average waiting time in a doctor鈥檚 office varies. The standard deviation of waiting times in a doctor鈥檚 office is 3.4 minutes. A random sample of 30 patients in the doctor鈥檚 office has a standard deviation of waiting times of 4.1 minutes. One doctor believes the variance of waiting times is greater than originally thought. What type of test should be used?

Short Answer

Expert verified
Use a Chi-Square test for variance.

Step by step solution

01

Identify the Parameter and Hypothesis

The problem involves assessing whether the variance of waiting times is greater than originally expected based on a sample. This calls for a hypothesis test about a population variance. We define the null hypothesis as \( H_0: \sigma^2 = 3.4^2 \), where \( \sigma^2 \) is the population variance, and the alternative hypothesis as \( H_a: \sigma^2 > 3.4^2 \), suggesting an increase.
02

Determine the Test Type

Testing for a population variance involves using a Chi-Square test. The Chi-Square test fits our situation where we have a sample variance and we want to infer about the population variance based on this sample.
03

Set Up the Test Statistic

The test statistic for a Chi-Square test on variance is calculated as \( \chi^2 = \frac{(n-1)s^2}{\sigma_0^2} \), where \( n \) is the sample size, \( s^2 \) is the sample variance, and \( \sigma_0^2 \) is the hypothesized population variance under the null hypothesis.
04

Interpret the Results

Compute the test statistic and compare it against the critical value from the Chi-Square distribution table at the desired significance level for \( n-1 \) degrees of freedom. For a right-tailed test like ours (since \( H_a \) suggests an increase), reject \( H_0 \) if the test statistic exceeds the critical value.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Hypothesis Testing
Hypothesis testing is a fundamental aspect of statistics, used to determine whether a hypothesis about a population parameter is supported by sample data. In our context, the doctor wants to verify if the variance of waiting times is greater than initially expected. To systematically investigate this claim, we use hypothesis testing. Here's a simple breakdown:
  • Null Hypothesis (): This is the default assumption. In our problem, it states that the population variance () remains at 3.4 minutes squared. Symbolically, it is expressed as \( H_0: 2 = 3.4^2 \).
  • Alternative Hypothesis (): This reflects the assertion we want to test. Here, it proposes that the population variance is greater than 3.4 minutes squared, denoted by \( H_a: 2 > 3.4^2 \).
The process then involves calculating a test statistic based on sample data, and deciding whether to accept or reject the null hypothesis based on this statistic. Understanding the logic of hypothesis testing simplifies decision-making by quantifying uncertainty and providing a calculated basis for claims.
Population Variance
Population variance () is a measure of how much individuals in a population differ from the population mean. It is calculated by determining the average of the squared deviations from the mean. To visualize:
  • Imagine a doctors waiting room where the time each patient waits varies. The population variance tells us how spread out these wait times are among all patients.
  • In our example, the expected population variance was calculated from historical data or expert assumption, set at \( 3.4^2 \). This acts as a standard to compare against the current sample.
When testing involves variance, it's crucial to determine if the sample data suggest a larger variance than what we assumed for the population. If so, this could indicate a change in the underlying process affecting waiting times, prompting further investigation.
Sample Variance
Sample variance () is similar to population variance, but it's derived from a smaller, randomly chosen subset from the larger population. It provides an estimate of the population variance. Lets simplify further:
  • Sample Variance Calculation: Measure how much each wait time within our sample deviates from its own mean, square these differences, and then average them, adjusting for the sample size.
  • In our situation, a sample of 30 patients yielded a sample variance based on a standard deviation of 4.1 minutes. The task is to determine if this suggests our population variance assumption is incorrect.
Sample variance is critical in our test since it serves as a bridge to infer about the population variance. It allows us to use the Chi-Square test to statistically substantiate whether the variation in wait times has indeed increased beyond the baseline assumption.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Isabella, an accomplished Bay to Breakers runner, claims that the standard deviation for her time to run the 7.5 mile race is at most three minutes. To test her claim, Rupinder looks up five of her race times. They are 55 minutes, 61 minutes, 58 minutes, 63 minutes, and 57 minutes.

Which test would you use to decide whether two factors have a relationship?

When looking at energy consumption, we are often interested in detecting trends over time and how they correlate among different countries. The information in Table 11.57 shows the average energy use (in units of kg of oil equivalent per capita) in the USA and the joint European Union countries (EU) for the six-year period 2005 to 2010. Do the energy use values in these two areas come from the same distribution? Perform the analysis at the 5% significance level.

Suppose an airline claims that its flights are consistently on time with an average delay of at most 15 minutes. It claims that the average delay is so consistent that the variance is no more than 150 minutes. Doubting the consistency part of the claim, a disgruntled traveler calculates the delays for his next 25 flights. The average delay for those 25 flights is 22 minutes with a standard deviation of 15 minutes. Is the traveler disputing the claim about the average or about the variance?

If the number of degrees of freedom for a chi-square distribution is 25, what is the population mean and standard deviation?
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.