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Chapter 8: Problem 62

When comparing two sample proportions with a two-sided alternative hypothesis, all other factors being equal, will you get a smaller p-value with a larger sample size or a smaller sample size? Explain.

Short Answer

Expert verified
A larger sample size would generally lead to a smaller p-value, all other factors being equal. This is due to the increased power of the test with larger samples which allows smaller differences to be recognized as statistically significant.

Step by step solution

01

Understand the P-value

A p-value is a measure of the strength of evidence in support of a null hypothesis. The smaller the p-value, the stronger the evidence that the null hypothesis should be rejected.
02

Understand the Sample Size

Sample size refers to the number of observations or replicates (the rows in your dataset). The larger the sample size, the more information you have and the closer you are to understanding the 'true' nature of the population.
03

Relation between P-value and Sample Size

With an increase in sample size, there's more power to detect the true effect in the population. Thus, with larger sample sizes, smaller differences are recognized as statistically significant, leading to smaller p-values.

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Key Concepts

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

P-Value
The p-value is a number that helps us decide the strength of results from our statistical tests. It tells us how likely it is to observe the data we have if the null hypothesis were true. Here's a simple way to look at it:
  • A small p-value (typically 鈮� 0.05) suggests that the observed data is not consistent with the assumption made by the null hypothesis.
  • A large p-value (> 0.05) suggests that the observed data is consistent with the null hypothesis.
The p-value does not tell us how big the effect is or even how important it is. It simply comments on the rarity of the data if the null hypothesis is true. Remember, a small p-value indicates strong evidence against the null hypothesis, suggesting a significant result.
Sample Size
Sample size refers to how many elements or participants are used when conducting a study or experiment. A larger sample size gives a clearer, more reliable picture of the population's characteristics. Consider these advantages of larger sample sizes:
  • Increased precision: Larger samples tend to reflect the population more accurately.
  • Reduced variability: There's lower risk of observing extreme outcomes just by chance.
  • Increased power: The ability to detect a true effect improves with more data. Understanding these benefits showcases why a larger sample would likely result in more trustworthy statistical outcomes, including smaller p-values if the effect truly exists.
Statistical Significance
Statistical significance determines whether the results of a study are likely to be true and not due to random chance. When we say a result is statistically significant, it simply means it has a low probability of happening under the null hypothesis.
  • A statistically significant result often comes with a p-value less than 0.05.
  • This threshold is not strict; different fields or studies might use 0.01 or 0.10.
Significance does not measure the size or importance of an effect. Instead, it deals with the confidence we have in the results not being due to random chance alone.
Null Hypothesis
The null hypothesis is a statement used in statistics to propose that there is no effect or no difference between groups or variables. It often forms the backbone of statistical testing.
  • The null hypothesis is usually the default or "no change" scenario.
  • It is the hypothesis that researchers typically seek to test against or reject.
An important part of hypothesis testing is often hoping to reject the null hypothesis in favor of an alternative hypothesis, which implies evidence for a significant effect or difference. However, failing to reject the null hypothesis does not mean it is true; it simply means there's not enough evidence against it given the data.

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Most popular questions from this chapter

A magazine advertisement claims that wearing a magnetized bracelet will reduce arthritis pain in those who suffer from arthritis. A medical researcher tests this claim with 233 arthritis sufferers randomly assigned either to wear a magnetized bracelet or to wear a placebo bracelet. The researcher records the proportion of each group who report relief from arthritis pain after 6 weeks. After analyzing the data, he fails to reject the null hypothesis. Which of the following are valid interpretations of his findings? There may be more than one correct answer. a. The magnetized bracelets are not effective at reducing arthritis pain. b. There's insufficient evidence that the magnetized bracelets are effective at reducing arthritis pain. c. The magnetized bracelets had exactly the same effect as the placebo in reducing arthritis pain. d. There were no statistically significant differences between the magnetized bracelets and the placebos in reducing arthritis pain.

Historically (from about 2001 to 2014 ), \(57 \%\) of Americans believed that global warming is caused by human activities. A March 2017 Gallup poll of a random sample of 1018 Americans found that 692 believed that global warming is caused by human activities. a. What percentage of the sample believed global warming was caused by human activities? b. Test the hypothesis that the proportion of Americans who believe global warming is caused by human activities has changed from the historical value of \(57 \%\). Use a significance level of \(0.01\). c. Choose the correct interpretation: i. In 2017 , the percentage of Americans who believe global warming is caused by human activities is not significantly different from \(57 \%\). ii. In 2017 , the percentage of Americans who believe global warming is caused by human activities has changed from the historical level of \(57 \%\).

Suppose you wanted to test the claim that the majority of U.S. voters are satisfied with the government response to the opioid crisis. State the null and alternative hypotheses you would use in both words and symbols.

Samuel Morse determined that the percentage of \(a\) 's in the English language in the 1800 s was \(8 \%\). A random sample of 600 letters from a current newspaper contained 60 a's. Using the \(0.10\) level of significance, test the hypothesis that the proportion of \(a\) 's in this modern newspaper is \(0.09\).

Choose one of the answers given. The null hypothesis is always a statement about a (sample statistic or population parameter).
See all solutions

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