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Chapter 9: Problem 13

Briefly explain the procedure used to calculate the \(p\) -value for a two- tailed and for a one-tailed test, respectively.

Short Answer

Expert verified
The p-value is calculated by determining the z-score for the observed results, then finding the probability of a result of this extreme or more so. For a two-tailed test, take the smaller probability of the two sides and multiply it by 2. For a one-tailed test, you only consider the possibility in one direction, thus no multiplication by 2 is needed.

Step by step solution

01

Concept of P-value

First, it's important to grasp what a p-value is. The p-value is the smallest level of significance at which we would reject the null hypothesis. It is used to determine the statistical significance of the observed results.
02

Calculation of the P-value in a Two-tailed test

In a two-tailed test, the null hypothesis is rejected if the observed results are significantly higher or lower than the expected results. To calculate the p-value, find the z-score (standard score) for the observed results. Then, calculate the probability of getting this result or a more extreme one from the z-distribution. Take the smaller side out of these two probabilities and multiply it by 2 to get the p-value.
03

Calculation of the P-value in a One-tailed test

In a one-tailed test, the null hypothesis is rejected only if the observed results are significantly higher (or lower, depending on the hypothesis). First, calculate the z-score for the observed results. Then, calculate the probability of getting this result or a more extreme one from the z-distribution. Unlike the two-tailed test, do not multiply by 2. This gives the p-value.

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Key Concepts

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

Understanding the Two-Tailed Test
A two-tailed test is a statistical test used to determine if a sample mean is significantly higher or lower than the population mean. In simple terms, it's checking both extremes. It's as if you're betting on both sides of an outcome.

Why would you use a two-tailed test? If you believe that there is a difference but aren鈥檛 sure about the direction, a two-tailed test helps with that. For example, if testing a new drug, you want to know if it's either more or less effective than the standard treatment.

Here's how it works:
  • You establish the null hypothesis (no effect or difference) and the alternative hypothesis (there is an effect or difference).
  • Using your data, you calculate a test statistic (like a z-score) that tells you how far data points are from the mean under the null hypothesis.
  • Find the probability (p-value) of observing a statistic as extreme as, or more so than, the observed statistic. In a two-tailed test, you multiply the smaller of the two tail probabilities by 2.
This method is comprehensive and doesn't assume a direction of the effect, giving a balanced view.
Exploring the One-Tailed Test
A one-tailed test tests for the possibility of the relationship in one direction only. It's simpler than a two-tailed test and is used when we have a specific direction of interest in the hypothesis.

Picture this: you expect a new teaching method to improve students' performance. You're only interested in whether it performs better, not worse, hence a one-tailed test is appropriate.

Steps to perform a one-tailed test include:
  • Set your null hypothesis (no change expected) and your alternative hypothesis (expecting a change in one direction).
  • Calculate the appropriate test statistic.
  • Determine the probability (p-value) of observing a test statistic as extreme as, or more extreme than, the one calculated, only in the direction of interest.
Since the direction is known, this method is more powerful but must be used with caution to ensure the hypothesis rightly matches the testing direction.
Defining Statistical Significance
Statistical significance is a way of determining whether a result from data collected in a study is likely to be due to chance or to a specific factor of interest.

Imagine flipping a coin and getting heads nine times in a row. If your hypothesis states that the coin is fair, this result is statistically suspicious. Statistical significance helps us decide if the observed outcome is rare enough to reject the null hypothesis.

Here鈥檚 a breakdown:
  • We choose a significance level (often 0.05). This is the threshold for decision-making.
  • We calculate the p-value - the probability of observing the effect when, in fact, no effect exists.
  • If the p-value is less than the significance level, the results are deemed statistically significant, leading to rejection of the null hypothesis.
Understanding statistical significance helps ensure we make correct inferences based on empirical data rather than random noise.
The Importance of the Null Hypothesis
The null hypothesis is a fundamental concept in statistics that serves as a starting point for statistical testing. It's the assumption that there is no effect or no difference.

Imagine you have a new method of teaching and you think it's better than the old one. The null hypothesis would state that there is no difference between the two methods in terms of effectiveness.

Here鈥檚 the process around the null hypothesis:
  • Formulate the null hypothesis. For many tests, this hypothesis posits no effect or no change.
  • Collect your data to test against this hypothesis.
  • Using statistical tests, determine the p-value. When it's less than the critical significance level, reject the null hypothesis in favor of the alternative.
The null hypothesis acts like a default scenario that you need strong evidence against, ensuring that researchers remain cautious about believing new findings without firm statistical backing.

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Most popular questions from this chapter

Lazurus Steel Corporation produces iron rods that are supposed to be 36 inches long. The machine that makes these rods does not produce each rod exactly 36 inches long. The lengths of the rods are approximately normally distributed and vary slightly. It is known that when the machine is working properly, the mean length of the rods is 36 inches. The standard deviation of the lengths of all rods produced on this machine is always equal to \(.035\) inch. The quality control department at the company takes a sample of 20 such rods every week, calculates the mean length of these rods, and tests the null hypothesis, \(\mu=36\) inches, against the alternative hypothesis, \(\mu \neq 36\) inches. If the null hypothesis is rejected, the machine is stopped and adjusted. A recent sample of 20 rods produced a mean length of \(36.015\) inches. a. Calculate the \(p\) -value for this test of hypothesis. Based on this \(p\) -value, will the quality control inspector decide to stop the machine and adjust it if he chooses the maximum probability of a Type I error to be .02? What if the maximum probability of a Type I error is \(10 ?\) b. Test the hypothesis of part a using the critical-value approach and \(\alpha=.02\). Does the machine need to be adjusted? What if \(\alpha=.10 ?\)

The manager of a restaurant in a large city claims that waiters working in all restaurants in his city earn an average of \(\$ 150\) or more in tips per week. A random sample of 25 waiters selected from restaurants of this city yielded a mean of \(\$ 139\) in tips per week with a standard deviation of \(\$ 28\). Assume that the weekly tips for all waiters in this city have a normal distribution. a. Using a \(1 \%\) significance level, can you conclude that the manager's claim is true? Use both approaches. b. What is the Type I error in this exercise? Explain. What is the probability of making such an error?

\(\quad\) Consider \(H_{0}: p=.70\) versus \(H_{1}: p \neq .70\). a. A random sample of 600 observations produced a sample proportion equal to .68. Using \(\alpha=.01\), would you reject the null hypothesis? b. Another random sample of 600 observations taken from the same population produced a sample proportion equal to \(.76\). Using \(a=.01\), would you reject the null hypothesis? Comment on the results of parts a and b.

According to an estimate, 2 years ago the average age of all CEOs of medium- sized companies in the United States was 58 years. Jennifer wants to check if this is still true. She took a random sample of 70 such CEOs and found their mean age to be 55 years with a standard deviation of 6 years. a. Suppose that the probability of making a Type I error is selected to be zero. Can you conclude that the current mean age of all CEOs of medium-sized companies in the United States is different from 58 years? b. Using a \(1 \%\) significance level, can you conclude that the current mean age of all CEOs of medium-sized companies in the United States is different from 58 years? Use both approaches.

The mean consumption of water per household in a city was 1245 cubic feet per month. Due to a water shortage because of a drought, the city council campaigned for water use conservation by households. A few months after the campaign was started, the mean consumption of water for a sample of 100 households was found to be 1175 cubic feet per month. The population standard deviation is given to be 250 cubic feet. a. Find the \(p\) -value for the hypothesis test that the mean consumption of water per household has decreased due to the campaign by the city council. Would you reject the null hypothesis at \(\alpha=.025\) ? b. Make the test of part a using the critical-value approach and \(\alpha=.025\).
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