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Chapter 10: Problem 14

A February 2007 story on www .redandblack.com stated, "Students who use laptops in class have lower GPAs, according to a study by Cornell University." a. Suppose this conclusion was based on a significance test comparing means. Defining notation in context, identify the groups and the population means and state the null hypothesis for the test. b. Suppose the study conclusion was based on a P-value of 0.01 obtained for the significance test mentioned in part a. Explain what you could learn from a confidence interval comparing the means that you are not able to learn from this P-value.

Short Answer

Expert verified
The groups are students using laptops vs. not using laptops, with \( \mu_1 \) and \( \mu_2 \) as their means. \( H_0: \mu_1 = \mu_2 \). A confidence interval gives the size of the difference, unlike the P-value.

Step by step solution

01

Define Groups and Population Means

Identify the groups involved in the study. **Group 1** is 'students using laptops in class' and their population mean GPA is denoted as \( \mu_1 \). **Group 2** is 'students not using laptops in class' and their population mean GPA is denoted as \( \mu_2 \).
02

State the Null Hypothesis

The null hypothesis (\( H_0 \)) for a significance test comparing means usually states that there is no difference between the two population means. Formally, \( H_0: \mu_1 = \mu_2 \).
03

Understand the Role of P-value

A P-value is used to determine the strength of the evidence against the null hypothesis. A P-value of 0.01 indicates strong evidence against the null hypothesis, suggesting a significant difference between the GPAs of the two groups.
04

Role of Confidence Interval

A confidence interval provides an estimate range for the difference in population means. Beyond indicating whether there is a significant difference, it shows the possible size and direction of that difference, which the P-value alone cannot reveal.
05

Comparison of P-value and Confidence Interval

While the P-value tells us whether there is enough evidence to reject the null hypothesis, the confidence interval gives us a range of plausible values for \( \mu_1 - \mu_2 \), giving insights into how much higher or lower one group's mean GPA might be compared to the other.

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Key Concepts

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

Confidence Interval
Confidence intervals are essential when we want to understand the range in which a population parameter lies, such as the difference between two population means. When you perform a study, getting a single numerical result is often not enough if you're looking at data that can vary. Instead, it's helpful to know an interval within which the true result likely falls. This is where confidence intervals come into play.
  • A confidence interval gives us a range, not just a point estimate. This can provide more comfort and assurance about the population parameter.
  • The level of confidence, often 95% or 99%, tells us how sure we are that the interval contains the true population mean difference.
  • In the context of comparing GPAs between students who use laptops and those who don't, the confidence interval will not only confirm if there is a difference but also demonstrate how large or small it might be.
Understanding a confidence interval helps in knowing the effect size, which allows schools or researchers to gauge if the difference is practically significant or just merely statistically significant.
P-Value
The P-value is a critical component in the process of hypothesis testing. It helps us assess the evidence against the null hypothesis. Specifically, in this study it was mentioned that a P-value of 0.01 was obtained.
  • A P-value quantitatively tells us how likely our results are, given the null hypothesis is true. A value of 0.01 indicates there's only a 1% probability that such a result, or a more extreme one, would occur by random chance, assuming the null hypothesis of equal population means is true.
  • Such a low P-value, like 0.01, suggests strong evidence against the null hypothesis. In the study about GPAs and laptop usage, this means we have strong evidence that the GPA difference is real and not just due to variability in the data.
  • However, the P-value does not tell us the size or direction of the difference, which can be provided by examining the confidence interval.
Thus, while a P-value is essential for determining statistical significance, it doesn't tell the complete story about the data you are analyzing.
Population Mean
When analyzing data, we often talk about population means, which represent the average value of a characteristic within a whole population. In this exercise, we are looking at two different populations: students using laptops and those not using them.
  • The population mean symbol, usually denoted as \( \mu \), refers to the average GPA for each group. It's crucial to understand these averages when looking for differences in behavior or outcomes across groups.
  • In hypothesis testing, we compare these population means to see if any observed differences are statistically significant. For example, when the null hypothesis \( H_0: \mu_1 = \mu_2 \) is proposed, it assumes no difference in average GPAs between the two groups.
  • Understanding what a population mean is and how it relates to each group helps solidify our understanding of how studies like this one interpret differences in academic performance.
Recognizing how your data set's population mean can sway or confirm hypotheses makes it a foundational concept in statistical analysis.

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Most popular questions from this chapter

The National Health Interview Survey conducted of 27,603 adults by the U.S. National Center for Health Statistics in 2009 indicated that \(20.6 \%\) of adults were current smokers. A similar study conducted in 1991 of 42,000 adults indicated that \(25.6 \%\) were current smokers. a. Find and interpret a point estimate of the difference between the proportion of current smokers in 1991 and the proportion of current smokers in 2009 . b. A \(99 \%\) confidence interval for the true difference is \((0.042,0.058) .\) Interpret. c. What assumptions must you make for the interval in part b to be valid?

TV watching A researcher predicts that the percentage of people who do not watch TV is higher now than before the advent of the Internet. Let \(p_{1}\) denote the population proportion of American adults in 1975 who reported watching no TV. Let \(p_{2}\) denote the corresponding population proportion in 2008 . a. Set up null and alternative hypotheses to test the researcher's prediction. b. According to General Social Surveys, 57 of the 1483 subjects sampled in 1975 and 87 of the 1324 subjects sampled in 2008 reported watching no TV. Find the sample estimates of \(p_{1}\) and \(p_{2}\). c. Show steps of a significance test. Explain whether the results support the reseacher's claim.

A study in Rotterdam (European Journal of Heart Failure, vol. \(11,2009,\) pp. \(922-928\) ) followed the health of 7983 subjects over 20 years to determine whether intake of fish could be associated with a decreased risk of heart failure in a general population of men and women aged 55 years and older. Results showed that the dietary intake of fish was not significantly related to heart failure incidence. Even for a high daily fish consumption of more than 20 grams a day there appeared no added protection against heart failure. Incidence rates were similar in those who consumed no fish (incidence rate of 11 per 1000 ), moderate fish (median \(9 \mathrm{~g}\) per day, 12.3 per 1000 ) or high fish ( 9.9 per 1000 ). The relative risk of heart failure in the high intake groups (medium and high fish) was 0.96 when compared with no intake with a \(95 \%\) confidence interval of \((0.78,1.18) .\) Explain how to interpret the (a) reported relative risk and (b) confidence interval for the relative risk.

Sampling sleep The 2009 Sleep in America poll of a random sample of 1000 adults reported that respondents slept an average of 6.7 hours on weekdays and 7.1 hours on weekends, and that \(28 \%\) of respondents got eight or more hours of sleep on weekdays whereas \(44 \%\) got eight or more hours of sleep on weekends (www.sleepfoundation.org). a. To compare the means or the percentages using inferential methods, should you treat the samples on weekdays and weekends as independent samples, or as dependent samples? Explain. b. To compare these results to polls of other people taken in previous years, should you treat the samples in the two years as independent samples, or as dependent samples? Explain.

Student survey \(\quad\) Refer to the FL Student Survey data file on the text CD. Use the number of times reading a newspaper as the response variable and gender as the explanatory variable. The observations are as follows: $$ \begin{array}{ll} \text { Females: } & 5,3,6,3,7,1,1,3,0,4,7,2,2,7,3,0,5,0,4,4, \\ & 5,14,3,1,2,1,7,2,5,3,7 \\ \text { Males: } & 0,3,7,4,3,2,1,12,1,6,2,2,7,7,5,3,14,3,7, \\ & 6,5,5,2,3,5,5,2,3,3 \end{array} $$ Using software, a. Construct and interpret a plot comparing responses by females and males. b. Construct and interpret a \(95 \%\) confidence interval comparing population means for females and males. c. Show all five steps of a significance test comparing the population means. d. State and check the assumptions for part b and part \(c\).
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