/*! 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 31 GPAs (Example 11) In finding a c... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 31

GPAs (Example 11) In finding a confidence interval for a random sample of 30 students GPAs, one interval was \((2.60,3.20)\) and the other was \((2.65,3.15)\). a. One of them is a \(95 \%\) interval and one is a \(90 \%\) interval. Which is which, and how do you know? b. If we used a larger sample size \((n=120\) instead of \(n=30\) ). would the \(95 \%\) interval be wider or narrower than the one reported here?

Short Answer

Expert verified
The confidence interval (2.60,3.20) is 95% interval and the confidence interval (2.65,3.15) is 90% interval. If the sample size increases from 30 to 120, the 95% confidence interval would be narrower than the ones provided.

Step by step solution

01

Identify confidence intervals

The given two intervals are (2.60,3.20) and (2.65,3.15), confidence interval with greater range should be associated with higher degree of confidence as greater uncertainty is involved. Therefore, (2.60,3.20) is likely to be the 95% interval and (2.65,3.15) is the 90% interval.
02

Effect of sample size on confidence interval

With increase in sample size, the confidence interval becomes narrower assuming the confidence level remains the same. This is due to the inverse relationship between sample size and standard error (precisely, standard deviation divided by square root of sample size, which decreases with increase in sample size). Therefore, if we used a larger sample size (n=120 instead of n=30), the 95% interval would be narrower than the one reported here.

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.

Sample Size Effect
When we collect data to make inferences about a population, the sample size plays a critical role in determining the accuracy of our estimates. A larger sample size generally means a more precise estimate of the population parameters.

Here's why:
  • As the sample size increases, the estimates we make tend to become more reliable because we capture more information about the population.
  • With a larger sample, we reduce the variability of our estimate, leading to a narrower confidence interval.
In the provided example, increasing the sample size from 30 students to 120 would result in a narrower confidence interval at the same confidence level. This is because the standard error decreases as the sample size increases. Therefore, a larger sample allows for a more precise range around the population mean.
Standard Error
The concept of Standard Error is key to understanding how confidence intervals shrink or expand. It represents the average distance that the sample mean of your data will be from the actual population mean.

The standard error can be calculated using the formula:\[SE = \frac{\sigma}{\sqrt{n}}\]where \(\sigma\) is the population standard deviation, and \(n\) is the sample size. As you can see, the standard error inversely relates to the sample size:
  • Larger samples produce a smaller standard error, which in turn leads to a narrower confidence interval.
  • This is because larger samples better approximate the population, reducing the influence of random sampling variability.
The smaller the standard error, the more our sample mean is tightly clustered around the true population mean, making our confidence intervals more precise.
95% vs. 90% Confidence Levels
Confidence level indicates how sure we are that the true population parameter lies within the confidence interval we calculated. Higher confidence levels mean wider intervals, capturing more potential variability in the sample data.

In our example:
  • The interval (2.60, 3.20) is wider, hence it is the 95% confidence interval. This suggests a higher degree of certainty but also more room to account for potential data variability.
  • The interval (2.65, 3.15) is narrower, likely corresponding to the 90% confidence interval, which implies less certainty but a more precise estimate.
Choosing between these intervals depends on the researcher鈥檚 willingness to trade precision for certainty. A 95% interval suggests we are very confident the true mean is within that range, while a 90% interval offers a slightly less certain, yet more precise, range.

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

Carrots The weights of four randomly chosen bags of horse carrots, each bag labeled 20 pounds, were \(20.5,19.8,20.8\), and \(20.0\) pounds. Assume that the distribution of weights is Normal. Find a \(95 \%\) confidence interval for the mean weight of all bags of horse carrots. Use technology for your calculations. a. Decide whether each of the following three statements is a correctly worded interpretation of the confidence interval, and fill in the blanks for the correct option(s). i. \(95 \%\) of all sample means based on samples of the same size will be between and ii. I am \(95 \%\) confident that the population mean is between and iii. We are \(95 \%\) confident that the boundaries are and b. Can you reject a population mean of 20 pounds? Explain.

Jacqueline Loya, a statistics student, asked students with jobs how many times they went out to eat in the last week. There were 25 students who had part- time jobs and 25 students who had full-time jobs. Carry out a hypothesis test to determine whether the mean number of meals out per week for students with full-time jobs is greater than that for those with part-time jobs. Use a significance level of \(0.05 .\) Assume that the conditions for a two-sample \(t\) -test hold. Full-time jobs: \(5,3,4,4,4,2,1,5,6,5,6,3,3,2,4,5,2,3,7,5\), \(5,1,4,6,7\) Part-time jobs: \(1,1,5,1,4,2,2,3,3,2,3,2,4,2,1,2,3,2,1,3\), \(3,2,4,2,1\)

Atkins Diet Difference Ten randomly selected people went on a Atkins diet for a month. The weight losses experienced (in pounds) were $$ 3,8,10,0,4,6,6,4,2, \text { and }-2 $$ The negative weight loss is a weight gain. Test the hypothesis that the mean weight loss was more than 0 , using a significance level of \(0.05\). Assume the population distribution is

Independent or Paired (Example 13) State whether each situation has independent or paired (dependent) samples. a. A researcher wants to know whether pulse rates of people go down after brief meditation. She collects the pulse rates of a random sample of people before meditation and then collects their pulse rates after meditation. b. A researcher wants to know whether women who text send more text messages than men who text. She gathers two random samples, one from men and one from women, and asks them how many text messages they sent yesterday.

Choose a test for each situation: one-sample \(t\) -test, twosample \(t\) -test, paired \(t\) -test, and no \(t\) -test. a. A researcher goes to a clothing store and observes whether each person is male or female and whether they return the clothes to the correct racks (yes or no) after trying them on. b. A random sample of restaurants is obtained. Then the researcher walks into each restaurant wearing ordinary clothes and finds out how long it takes (in minutes) for a waiter to approach the researcher. Later, the researcher goes into the same restaurant dressed in expensive clothes and finds out how long it takes for a waiter to approach. c. The supervisor observes the number of working hours of a random sample of part-time workers and a random sample of regular workers.
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Pure 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.