/*! 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 39 Newspaper headlines recently ann... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 23: Problem 39

Newspaper headlines recently announced a decline in science scores among high school seniors. In \(2000,\) a total of 15,109 seniors tested by The National Assessment in Education Program (NAEP) scored a mean of 147 points. Four years earlier, 7537 seniors had averaged 150 points. The standard error of the difference in the mean scores for the two groups was 1.22 a) Have the science scores declined significantly? Cite appropriate statistical evidence to support your conclusion. b) The sample size in 2000 was almost double that in \(1996 .\) Does this make the results more convincing or less? Explain.

Short Answer

Expert verified
a) Yes, scores declined significantly; the test statistic exceeds the critical value. b) The larger sample size in 2000 makes the results more convincing.

Step by step solution

01

Define the null and alternative hypotheses

Our goal is to determine if there has been a significant decline in science scores. We establish our null hypothesis (H0) as there being no significant difference in mean scores between the two groups, i.e., \( \mu_1 - \mu_2 = 0 \). The alternative hypothesis (H1) is that there is a significant decline in scores, i.e., \( \mu_1 - \mu_2 < 0 \), where \( \mu_1 \) and \( \mu_2 \) are the mean scores for 1996 and 2000, respectively.
02

Calculate the test statistic

The test statistic for the difference in means can be calculated using the formula: \[ z = \frac{\text{mean difference}}{\text{standard error of difference}} = \frac{150 - 147}{1.22} \approx 2.46 \].
03

Determine the critical value and make the decision

Using a significance level of \( \alpha = 0.05 \) for a one-tailed test, a critical value from the z-table for \( \alpha = 0.05 \) is approximately 1.645. Since the calculated test statistic \( z = 2.46 \) is greater than 1.645, we reject the null hypothesis. This indicates a significant decline in scores.
04

Analyze the impact of sample size

A larger sample size in 2000 increases the reliability of its mean score, reducing the impact of random variation and providing more robust results. Thus, the larger sample size in 2000 makes the results more convincing because the mean estimate for 2000 is more precise.

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.

Null Hypothesis
In statistics, the null hypothesis is a statement that suggests there is no effect or no difference between certain phenomena or populations. It is represented as the baseline statement which researchers attempt to test against. For example, if we want to determine whether new science scores have changed, the null hypothesis (\(H_0\)) would state that there is no difference between the mean scores of two different years.
  • It provides a point of reference for statistically testing a claim.
  • It is usually formulated to be nullified or rejected based on evidence from data.
In the original exercise, the null hypothesis was that there was no significant difference in mean scores between 1996 and 2000 for high school seniors. This serves as the default assumption unless proven otherwise through statistical testing.
Alternative Hypothesis
Contrary to the null hypothesis, the alternative hypothesis suggests that there is an effect or a difference. It is what researchers aim to support through their data analysis. For the given problem, the alternative hypothesis (\(H_1\)) posits that there has been a decline in science scores from 1996 to 2000.
  • Typically states the presence of a relationship or the reality of change.
  • Serves as a counterclaim to the null hypothesis.
In the context of the example exercise, the alternative hypothesis suggests that the mean score in 2000 is less than that in 1996. This hypothesis gains support if the test statistic demonstrates a statistically significant result.
Test Statistic
A test statistic is a standardized value that is calculated from sample data during a hypothesis test. It is used to decide whether to reject the null hypothesis. A higher test statistic value typically indicates a greater divergence from the null hypothesis.
  • It compares your sample data against what the null hypothesis predicts.
  • Different tests have different ways of calculating this statistic.
In the exercise, the test statistic is derived from the formula:\[z = \frac{\text{mean difference}}{\text{standard error of difference}} = \frac{150 - 147}{1.22} \approx 2.46\]A critical value is then used to compare this score. Since the calculated z-score of 2.46 exceeds the critical z-value of 1.645 for a confidence level of 95%, it indicates a significant decline in scores, prompting rejection of the null hypothesis.
Sample Size
Sample size is the number of observations or replicates used in a study. It is an important aspect as it affects the precision of the statistical analysis and the power of the hypothesis test.
  • Larger sample sizes generally provide more reliable and valid results as they are less likely to be affected by random outliers or variations.
  • A small sample size may lead to less reliable results, potentially causing Type I or Type II errors.
In our scenario, increasing the number of students tested in 2000 makes the test results more convincing. This is due to a reduction in random sampling error, leading to a more accurate and stable representation of the population's mean score for that year. A larger sample size strengthens the overall conclusions drawn from the statistical test.

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 Consumer Reports article described in Exercise 1 also listed the sodium content (in mg) for the various hot dogs tested. A test of the null hypothesis that beef hot dogs and meat hot dogs don't differ in the mean amounts of sodium yields a P-value of \(0.11 .\) Would a \(95 \%\) confidence interval for \(\mu_{\text {Meat}}-\mu_{\text {Beef}}\) include \(0 ?\) Explain.
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Geometry

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.