Hunter Was Testing $H_0: \mu = 100$ Versus $H_a: \mu \ \textless \ 100$ With A Sample Of 50 Observations. His Sample Mean Was 103, And His Sample Standard Deviation Was 5. Assume That The Conditions For Inference Were Met.Which Of

by ADMIN 233 views

Introduction

In statistical hypothesis testing, researchers often seek to determine whether a sample of data provides sufficient evidence to reject a null hypothesis in favor of an alternative hypothesis. In this article, we will explore the process of hypothesis testing, focusing on a specific scenario involving a sample mean. We will examine the conditions for inference, calculate the test statistic, and determine the p-value to make a decision about the null hypothesis.

The Null and Alternative Hypotheses

In this scenario, Hunter is testing the following hypotheses:

  • Null Hypothesis (H0): μ = 100 (The population mean is equal to 100.)
  • Alternative Hypothesis (Ha): μ < 100 (The population mean is less than 100.)

The Sample Data

Hunter has collected a sample of 50 observations, with a sample mean (x̄) of 103 and a sample standard deviation (s) of 5. The conditions for inference have been met, which means that the sample is randomly selected from the population, and the sample size is sufficiently large.

Calculating the Test Statistic

To perform the hypothesis test, we need to calculate the test statistic. Since we are testing a hypothesis about the population mean, we will use the t-statistic. The formula for the t-statistic is:

t = (x̄ - μ) / (s / √n)

where x̄ is the sample mean, μ is the population mean, s is the sample standard deviation, and n is the sample size.

Plugging in the values, we get:

t = (103 - 100) / (5 / √50) = 3 / (5 / 7.071) = 3 / 0.7071 = 4.25

Determining the p-value

The p-value represents the probability of observing a test statistic at least as extreme as the one we obtained, assuming that the null hypothesis is true. To calculate the p-value, we need to determine the degrees of freedom (df) for the t-distribution. The df is calculated as:

df = n - 1 = 50 - 1 = 49

Using a t-distribution table or calculator, we find that the p-value associated with a t-statistic of 4.25 and 49 degrees of freedom is approximately 0.

Making a Decision

Since the p-value is less than 0.05, we reject the null hypothesis. This means that there is sufficient evidence to suggest that the population mean is less than 100.

Conclusion

In this article, we have explored the process of hypothesis testing, focusing on a scenario involving a sample mean. We calculated the test statistic, determined the p-value, and made a decision about the null hypothesis. By following these steps, researchers can use statistical hypothesis testing to make informed decisions about their data.

Key Takeaways

  • The null and alternative hypotheses are statements about the population parameter.
  • The sample data is used to calculate the test statistic.
  • The p-value represents the probability of observing a test statistic at least as extreme as the one we obtained, assuming that the null hypothesis is true.
  • If the p-value is less than the significance level (usually 0.05), we reject the null hypothesis.

References

  • Moore, D. S., & McCabe, G. P. (2012). Introduction to the practice of statistics. W.H. Freeman and Company.
  • Larson, R. E., & Farber, M. (2013). Elementary statistics: Picturing the world. Cengage Learning.

Further Reading

  • For a more detailed explanation of hypothesis testing, see Moore and McCabe (2012).
  • For a discussion of the conditions for inference, see Larson and Farber (2013).

Code

# Load the necessary libraries
library(tidyverse)

# Define the sample data
x_bar <- 103
mu <- 100
s <- 5
n <- 50

# Calculate the test statistic
t_stat <- (x_bar - mu) / (s / sqrt(n))

# Print the test statistic
print(t_stat)

# Calculate the p-value
p_value <- pt(t_stat, df = n - 1, lower.tail = FALSE)

# Print the p-value
print(p_value)

Q&A: Hypothesis Testing

Q: What is hypothesis testing?

A: Hypothesis testing is a statistical method used to determine whether a sample of data provides sufficient evidence to reject a null hypothesis in favor of an alternative hypothesis.

Q: What are the null and alternative hypotheses?

A: The null hypothesis (H0) is a statement about the population parameter that is assumed to be true, while the alternative hypothesis (Ha) is a statement about the population parameter that is being tested.

Q: What are the conditions for inference?

A: The conditions for inference are:

  1. Random sampling: The sample must be randomly selected from the population.
  2. Sufficient sample size: The sample size must be sufficiently large to ensure that the sample is representative of the population.
  3. Independence: The observations in the sample must be independent of each other.
  4. Normality: The population distribution must be normal or the sample size must be sufficiently large to ensure that the Central Limit Theorem holds.

Q: What is the t-statistic?

A: The t-statistic is a measure of the difference between the sample mean and the population mean, standardized by the sample standard deviation and the sample size.

Q: How is the t-statistic calculated?

A: The t-statistic is calculated using the following formula:

t = (x̄ - μ) / (s / √n)

where x̄ is the sample mean, μ is the population mean, s is the sample standard deviation, and n is the sample size.

Q: What is the p-value?

A: The p-value is the probability of observing a test statistic at least as extreme as the one we obtained, assuming that the null hypothesis is true.

Q: How is the p-value calculated?

A: The p-value is calculated using a t-distribution table or calculator, and is determined by the degrees of freedom (df) and the t-statistic.

Q: What is the significance level?

A: The significance level is the maximum probability of rejecting the null hypothesis when it is true. It is usually set at 0.05.

Q: What happens if the p-value is less than the significance level?

A: If the p-value is less than the significance level, we reject the null hypothesis. This means that there is sufficient evidence to suggest that the population parameter is not equal to the value specified in the null hypothesis.

Q: What happens if the p-value is greater than or equal to the significance level?

A: If the p-value is greater than or equal to the significance level, we fail to reject the null hypothesis. This means that there is not sufficient evidence to suggest that the population parameter is not equal to the value specified in the null hypothesis.

Q: What are some common mistakes to avoid when performing hypothesis testing?

A: Some common mistakes to avoid when performing hypothesis testing include:

  1. Failing to check the conditions for inference: Make sure that the sample is randomly selected, the sample size is sufficiently large, and the observations are independent.
  2. Using the wrong significance level: Make sure to use the correct significance level for the test.
  3. Interpreting the p-value incorrectly: Make sure to understand the meaning of the p-value and how it relates to the null hypothesis.
  4. Not considering the effect size: Make sure to consider the effect size of the test, which is the magnitude of the difference between the sample mean and the population mean.

Q: What are some common applications of hypothesis testing?

A: Some common applications of hypothesis testing include:

  1. Comparing the means of two or more groups: Hypothesis testing can be used to compare the means of two or more groups to determine if there are significant differences between them.
  2. Testing the effect of a treatment: Hypothesis testing can be used to test the effect of a treatment on a population.
  3. Determining the relationship between two variables: Hypothesis testing can be used to determine the relationship between two variables.

Q: What are some common tools and software used for hypothesis testing?

A: Some common tools and software used for hypothesis testing include:

  1. R: A programming language and environment for statistical computing and graphics.
  2. Python: A programming language and environment for statistical computing and graphics.
  3. SPSS: A statistical software package for data analysis and hypothesis testing.
  4. SAS: A statistical software package for data analysis and hypothesis testing.

Q: What are some common resources for learning about hypothesis testing?

A: Some common resources for learning about hypothesis testing include:

  1. Textbooks: There are many textbooks available on hypothesis testing, including "Introduction to the Practice of Statistics" by Moore and McCabe and "Elementary Statistics: Picturing the World" by Larson and Farber.
  2. Online courses: There are many online courses available on hypothesis testing, including courses on Coursera, edX, and Udemy.
  3. Research articles: There are many research articles available on hypothesis testing, including articles in journals such as the Journal of Statistical Software and the Journal of Educational and Behavioral Statistics.
  4. Online communities: There are many online communities available for learning about hypothesis testing, including communities on Reddit and Stack Overflow.