/*! 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 76 A consulting agency was asked by... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 76

A consulting agency was asked by a large insurance company to investigate if business majors were better salespersons than those with other majors. A sample of 20 salespersons with a business degree showed that they sold an average of 11 insurance policies per week. Another sample of 25 salespersons with a degree other than business showed that they sold an average of 9 insurance policies per week.

Short Answer

Expert verified
To determine if there's a significant difference in policy sales between the two groups, statistical tests like an independent samples t-test could be performed. This test however requires additional information like standard deviations of the groups. If the p-value obtained is less than the significance level (typically 0.05), we could reject the null hypothesis and conclude that business majors sell more policies per week on average.

Step by step solution

01

Understand the Variables

Firstly, we identify and understand the variables in this problem. We have two independent groups - the group of salespersons with business degrees, and the group of salespersons with other degrees - and their weekly sales of insurance policies is the dependent variable.
02

Calculate the Mean

The mean or average is given as 11 policies per week for business degree holders and 9 policies for non-business degree holders.
03

Hypothesis Testing

Traditional hypothesis testing involves a null hypothesis and an alternative hypothesis. The null hypothesis (H0) might be that there is no difference in the average number of policies sold between the two groups. The alternative hypothesis (Ha) would then be that there is a difference.
04

Sample size

The problem statement provides the sample sizes: n1 = 20 for salespersons with business degrees and n2 = 25 for those without.
05

Statistical Analysis

Given all the previous information, a statistical test (such as an independent samples t-test) can be performed to determine if there is a statistically significant difference between the two groups. Without the standard deviations, we assume equal variances which allows us to use a basic comparison of means test.
06

Interpretation

Once the statistical test has been performed, the result (the p-value) can be compared with a set significance level (usually 0.05). If the p-value is less than the significance level, then we can reject the null hypothesis in favour of the alternative hypothesis - that is, conclude that business majors sell more policies per week on average.

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.

Independent Samples t-Test
When you want to compare the averages from two independent groups, an independent samples t-test is a handy statistical tool. It helps determine if there is a significant difference between the means of two groups. Here, we look at business majors versus non-business majors to see who sells more insurance policies on average.

The test works by calculating the difference between the two groups' means and then assessing how likely it is that this difference occurred by chance. If the calculated t-value is larger than the critical value from the t-distribution, it suggests the observed difference is likely not due to chance.
  • Assumptions: It assumes that the data was collected randomly, the groups are independent, and the data follows a normal distribution.
  • Process: Gather data, state your hypotheses, calculate the t-statistic, and interpret the significance.
  • Importance: Provides a statistical method to make informed decisions based on data analysis.
Null Hypothesis
The null hypothesis is a starting point in hypothesis testing that assumes no effect or difference between the groups being compared. It's often denoted as \(H_0\). In our case, the null hypothesis would be that there is no difference in the number of insurance policies sold by business majors compared to non-business majors.

It's critical because it provides a baseline for the test. By assuming no difference, we need strong evidence (through data) to reject it. The goal is to see if the data presents enough evidence to cast doubt on the null hypothesis's truth.
  • Symbol: Typically denoted as \(H_0\).
  • Role: Acts as a default assumption that researchers seek to test against.
  • Outcome: Getting a p-value smaller than the significance level can result in rejecting \(H_0\).
Alternative Hypothesis
The alternative hypothesis stands in contrast to the null hypothesis and represents a new claim based on observed effects. It's commonly represented as \(H_a\) or \(H_1\). In this scenario, the alternative hypothesis would be that business majors sell more insurance policies per week than non-business majors.

The alternative hypothesis suggests that there is a statistically significant difference worth exploring further. It becomes the preferred hypothesis if the null hypothesis is rejected based on the statistical evidence.
  • Representation: Often symbolized by \(H_a\) or \(H_1\).
  • Purpose: Proposed when the data shows a strong deviation from the null hypothesis scenario.
  • Decision Making: If evidence supports it, the alternative hypothesis can lead to actionable conclusions.
Sample Size
Sample size is the number of observations in a subset of data taken from a larger population. It's crucial because it affects the accuracy and reliability of statistical tests. In our example, we have two sample sizes: 20 for business majors and 25 for non-business majors.

A larger sample size generally provides more reliable results because it better reflects the population. However, the sample size also needs to be manageable and practical to obtain.
  • Influence: Larger sample sizes give results that are generally considered more stable and reliable.
  • Calculation: Statisticians use methods to determine how large a sample should be to achieve desired confidence levels.
  • Consideration: Must balance between being large enough for reliability and small enough for feasibility.

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

Explain what conditions must hold true to use the \(t\) distribution to make a confidence interval and to test a hypothesis about \(\mu_{1}-\mu_{2}\) for two independent samples selected from two populations with unknown but equal standard deviations.

As mentioned in Exercise \(10.26\), a town that recently started a single-stream recycling program provided 60-gallon recycling bins to 25 randomly selected households and 75-gallon recycling bins to 22 randomly selected households. The average total volumes of recycling over a 10 -week period were 382 and 415 gallons for the two groups, respectively, with standard deviations of \(52.5\) and \(43.8\) gallons, respectively. Suppose that the standard deviations for the two populations are not equal. a. Construct a \(98 \%\) confidence interval for the difference in the mean volumes of 10 -week recyclying for the households with the 60 - and 75 -gallon bins. b. Using the \(2 \%\) significance level, can you conclude that the average 10 -week recycling volume of all households having 60 -gallon containers is different from the average 10-week recycling volume of all households that have 75 -gallon containers? c. Suppose that the sample standard deviations were \(59.3\) and \(33.8\) gallons, respectively. Redo parts a and b. Discuss any changes in the results.

The management of a supermarket chain wanted to investigate if the percentages of men and women who prefer to buy national brand products over the store brand products are different. A sample of 600 men shoppers at the company's supermarkets showed that 246 of them prefer to buy national brand products over the store brand products. Another sample of 700 women shoppers at the company's supermarkets showed that 266 of them prefer to buy national brand products over the store brand products. a. What is the point estimate of the difference between the two population proportions? b. Construct a \(95 \%\) confidence interval for the difference between the proportions of all men and all women shoppers at these supermarkets who prefer to buy national brand products over the store brand products. c. Testing at the \(5 \%\) significance level, can you conclude that the proportions of all men and all women shoppers at these supermarkets who prefer to buy national brand products over the store brand products are different?

The manager of a factory has devised a detailed plan for evacuating the building as quickly as possible in the event of a fire or other emergency. An industrial psychologist believes that workers actually leave the factory faster at closing time without following any system. The company holds fire drills periodically in which a bell sounds and workers leave the building according to the system. The evacuation time for each drill is recorded. For comparison, the psychologist also records the evacuation time when the bell sounds for closing time each day. A random sample of 36 fire drills showed a mean evacuation time of \(5.1\) minutes with a standard deviation of \(1.1\) minutes. A random sample of 37 days at closing time showed a mean evacuation time of \(4.2\) minutes with a standard deviation of \(1.0\) minute. a. Construct a \(99 \%\) confidence interval for the difference between the two population means. b. Test at the \(5 \%\) significance level whether the mean evacuation time is smaller at closing time than during fire drills.

The manufacturer of a gasoline additive claims that the use of this additive increases gasoline mileage. A random sample of six cars was selected, and these cars were driven for 1 week without the gasoline additive and then for 1 week with the gasoline additive. The following table gives the miles per gallon for these cars without and with the gasoline additive. $$ \begin{array}{l|cccccc} \hline \text { Without } & 24.6 & 28.3 & 18.9 & 23.7 & 15.4 & 29.5 \\ \hline \text { With } & 26.3 & 31.7 & 18.2 & 25.3 & 18.3 & 30.9 \\ \hline \end{array} $$ a. Construct a \(99 \%\) confidence interval for the mean \(\mu_{d}\) of the population paired differences, where a paired difference is equal to the miles per gallon without the gasoline additive minus the miles per gallon with the gasoline additive. b. Using the \(2.5 \%\) significance level, can you conclude that the use of the gasoline additive increases the gasoline mileage?
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

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.