/*! 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 26 A coffee manufacturer is interes... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 26

A coffee manufacturer is interested in whether the mean daily consumption of regular-coffee drinkers is less than that of decaffeinated-coffee drinkers. Assume the population standard deviation for those drinking regular coffee is 1.20 cups per day and 1.36 cups per day for those drinking decaffeinated coffee. A random sample of 50 regular-coffee drinkers showed a mean of 4.35 cups per day. A sample of 40 decaffeinated-coffee drinkers showed a mean of 5.84 cups per day. Use the .01 significance level. Compute the \(p\) -value.

Short Answer

Expert verified
The p-value is virtually 0, leading us to reject the null hypothesis.

Step by step solution

01

State the Hypotheses

Null Hypothesis, \(H_0\): The mean daily consumption of regular coffee drinkers is equal to that of decaffeinated coffee drinkers. Alternative Hypothesis, \(H_1\): The mean daily consumption of regular coffee drinkers is less than that of decaffeinated coffee drinkers.
02

Identify the Means and Standard Deviations

Regular coffee: \( \mu_R = 4.35\), \( \sigma_R = 1.20\), \(n_R = 50\). Decaffeinated coffee: \( \mu_D = 5.84\), \( \sigma_D = 1.36\), \(n_D = 40\).
03

Calculate the Standard Error of the Difference

The standard error (SE) is given by \[SE = \sqrt{\frac{\sigma_R^2}{n_R} + \frac{\sigma_D^2}{n_D}}\]Substitute the values:\[SE = \sqrt{\frac{1.20^2}{50} + \frac{1.36^2}{40}} = \sqrt{0.0288 + 0.04624} \approx 0.285\]
04

Compute the Test Statistic

The test statistic \(Z\) is calculated as: \[Z = \frac{\mu_R - \mu_D}{SE}\]Substitute the means and SE:\[Z = \frac{4.35 - 5.84}{0.285} \approx -5.23\]
05

Determine the p-value

Using a standard normal distribution table, find the \(p\)-value associated with \(Z = -5.23\). The \(p\)-value is virtually zero (far below the 0.01 significance level).
06

Make a Decision

Since the \(p\)-value is less than the significance level of 0.01, reject the null hypothesis. This suggests that the mean daily consumption of regular coffee drinkers is significantly less than that of decaffeinated coffee drinkers.

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.

Population Standard Deviation
The population standard deviation is a measure that quantifies the amount of variation or dispersion in a set of population data. It's an essential component in hypothesis testing as it gives insights into the variability within groups. For instance, in the coffee consumption example, the population standard deviation for regular coffee is 1.20 cups per day, while for decaffeinated coffee, it is 1.36 cups per day.
  • This implies that the daily coffee consumption among regular drinkers tends to be closer to its average than it is among decaffeinated drinkers.
  • Lower standard deviation signifies less variability, meaning most values hover near the mean.
  • Higher standard deviation indicates that the data points spread out over a wider range of values.
Understanding the population standard deviation is crucial as it helps in calculating other statistical measures, such as standard error and test statistics, ultimately contributing to the interpretation of hypothesis tests.
p-value
The p-value is a key concept in hypothesis testing. It quantifies the probability of observing a test statistic at least as extreme as the one actually observed, assuming that the null hypothesis is true. In simpler terms, it's the probability that the results of your sample data occur by random chance.
  • A low p-value (typically less than 0.05) suggests that the observed data are unlikely under the null hypothesis, leading us to reject the null hypothesis.
  • In the coffee consumption exercise, the calculated p-value is virtually zero, far below the threshold of 0.01.
  • This indicates strong evidence against the null hypothesis, supporting the claim that mean consumption differs between regular and decaf drinkers.
Thus, the p-value serves as a "measure of evidence" against the null hypothesis, allowing us to understand how likely our results are due to random variation alone.
Significance Level
The significance level, denoted by alpha (伪), is the probability of rejecting the null hypothesis when it is actually true. It represents the threshold or cut-off for deciding whether a p-value indicates a statistically significant result.
  • In hypothesis testing, this is often set at 0.05, but in our case, it is 0.01, indicating stricter criteria for rejecting the null hypothesis.
  • This means that we only accept a 1% risk of concluding that a difference exists when there is none.
  • If the p-value is less than the significance level, we reject the null hypothesis in favor of the alternative hypothesis.
Selecting a significance level involves a balance between Type I error (false positive) and Type II error (false negative). In rigorous fields, lower significance levels are preferred as they reduce the probability of falsely claiming a discovery.
Standard Error
The standard error (SE) is a measure of the statistical accuracy of an estimate, indicating how much sample means can be expected to differ from the true population mean. It is crucial in the context of hypothesis testing as it helps in calculating test statistics like the Z-score.
  • The formula for standard error in comparing two means is \[SE = \sqrt{\frac{\sigma_R^2}{n_R} + \frac{\sigma_D^2}{n_D}}\]
  • In our exercise, the calculated SE was approximately 0.285, reflecting the combined variability of both samples.
  • Standard error decreases as sample size increases, making estimates more precise.
Standard error plays a pivotal role as it allows us to quantify how variable our sample means are and determine the reliability of our difference between sample means through the test statistic.

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

The research department at the home office of New Hampshire Insurance conducts ongoing research on the causes of automobile accidents, the characteristics of the drivers, and so on. A random sample of 400 policies written on single persons revealed 120 had at least one accident in the previous three-year period. Similarly, a sample of 600 policies written on married persons revealed that 150 had been in at least one accident. At the .05 significance level, is there a significant difference in the proportions of single and married persons having an accident during a three-year period? Determine the \(p\) -value.

Gibbs Baby Food Company wishes to compare the weight gain of infants using its brand versus its competitor's. A sample of 40 babies using the Gibbs products revealed a mean weight gain of 7.6 pounds in the first three months after birth. For the Gibbs brand the population standard deviation of the sample is 2.3 pounds. A sample of 55 babies using the competitor's brand revealed a mean increase in weight of 8.1 pounds. The population standard deviation is 2.9 pounds. At the .05 significance level, can we conclude that babies using the Gibbs brand gained less weight? Compute the \(p\) value and interpret it.

A nationwide sample of influential Republicans and Democrats was asked as a part of a comprehensive survey whether they favored lowering environmental standards so that high-sulfur coal could be burned in coal-fired power plants. The results were: $$ \begin{array}{|lcc|} \hline & \text { Republicans } & \text { Democrats } \\ \hline \text { Number sampled } & 1,000 & 800 \\ \text { Number in favor } & 200 & 168 \end{array} $$ At the .02 level of significance, can we conclude that there is a larger proportion of Democrats in favor of lowering the standards? Determine the \(p\) -value.

Fry Brothers Heating and Air Conditioning, Inc. employs Larry Clark and George Murnen to make service calls to repair furnaces and air conditioning units in homes. Tom Fry, the owner, would like to know whether there is a difference in the mean number of service calls they make per day. Assume the population standard deviation for Larry Clark is 1.05 calls per day and 1.23 calls per day for George Murnen. A random sample of 40 days last year showed that Larry Clark made an average of 4.77 calls per day. For a sample of 50 days George Murnen made an average of 5.02 calls per day. At the .05 significance level, is there a difference in the mean number of calls per day between the two employees? What is the \(p\) -value?

A recent study compared the time spent together by single- and dual-earner couples. According to the records kept by the wives during the study, the mean amount of time spent together watching television among the single-earner couples was 61 minutes per day, with a standard deviation of 15.5 minutes. For the dual-earner couples, the mean number of minutes spent watching television was 48.4 minutes, with a standard deviation of 18.1 minutes. At the .01 significance level, can we conclude that the single-earner couples on average spend more time watching television together? There were 15 single-earner and 12 dual-earner couples studied.
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation

Statistics

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.