What Is The Null Hypothesis Tested By Analysis Of Variance ✓ Solved
What is the null hypothesis tested by analysis of variance?
Textbook 1: Lane et al. Introduction to Statistics, David M. Lane et al., 2013.
Textbook 2: Illowsky et al. Introductory Statistics, Barbara Illowsky et al., 2013.
1. What is the null hypothesis tested by analysis of variance?
5. What is the difference between “N” and “n”?
8. What kind of skew does the F distribution have?
10. Assume an experiment is conducted with 5 conditions and 6 subjects in each condition. What are dfnumerator and dfdenominator?
HINT: In order to create an ANOVA table, you will need to follow examples 13.1 and 13.2 in your book. At the end of this homework assignment, there is also another detailed example. DO NOT calculate the SSwithin by hand.
Use EXCEL (or calculate by hand) the value for SStotal and then calculate the value for SSbetween. The value of SSwithin = SStotal – SSbetween. Use the equations in the summary table at the top of page 699 to find the value of the F-statistic and then find the P-value.
Use the following information to answer #61 and #63. Suppose a group is interested in determining whether teenagers obtain their drivers licenses at approximately the same average age across the country.
Null-Hypothesis is H0: μ1 = μ2 = μ3 = μ4 = μ5; Alternate Hypothesis is Hα: At least any two of the group means μ1, μ2, …, μ5 are not equal.
61. Find the degrees of freedom (numerator) = df (num).
63. Find the F-statistic using an ANOVA table.
69. A researcher wants to know if the mean times (in minutes) that people watch their favorite news station are the same.
71. Are the mean number of times a month a person eats out the same for whites, blacks, Hispanics, and Asians?
77. A grassroots group opposed to a proposed increase in the gas tax claimed that the increase would hurt working-class people the most.
81. Is the variance for the amount of money, in dollars, that shoppers spend on Saturdays at the mall the same as the variance for the amount of money that shoppers spend on Sundays at the mall?
HINT: See Section 13.4 (Test of Two Variances) to find the F-statistic.
Assume the null hypothesis is the variances are the same and assume the alternate hypothesis is.
Example: Calculations in the Analysis of Variance (ANOVA) Example Solution Table P-value using F-distribution (use EXCEL formula) FDIST (F, df1, df2) = FDIST(9.085,4,45) = 0.
Paper For Above Instructions
The analysis of variance (ANOVA) is a powerful statistical method used to compare means among multiple groups, thereby determining if any significant differences exist among them. Central to ANOVA is the null hypothesis, which usually posits that all group means are equal. In the context of this assignment, the null hypothesis tested by ANOVA is denoted as H0: μ1 = μ2 = μ3 = μ4 = μ5, which implies that there is no variation in the ages at which teenagers across different regions obtain their drivers licenses.
The distinction between “N” and “n” is crucial in statistics. “N” refers to the total population size, while “n” indicates the sample size drawn from that population. For instance, if we survey 30 teenagers from a larger population of 1,000, then N = 1,000 and n = 30. This distinction is vital when conducting ANOVA, as it helps in understanding the generalizability of the results.
The F distribution, used prominently in ANOVA, is known for its right-skewed nature. This skewness arises because the values it can take are non-negative, reflecting that variances, which are squared values, cannot be negative. As the degrees of freedom increase, the shape of the F distribution approaches a normal distribution, which is critical for determining critical values in hypothesis testing.
To calculate the degrees of freedom for an ANOVA model, we look at the numerator and denominator separately. The degrees of freedom for the numerator (dfn) are determined by the number of groups minus one (k - 1), where k is the total number of groups. Conversely, the degrees of freedom for the denominator (dfd) are calculated as the total number of observations minus the number of groups (N - k). In your case, with 5 conditions and 6 subjects each, dfn = 5 - 1 = 4, and dfd = 30 - 5 = 25.
Next, we proceed to calculate SStotal and SSbetween using Excel. SStotal represents the overall variability in the data, while SSbetween captures the variability attributed to the differences between group means. The formula for SStotal can be calculated by summing the squared deviations of each observation from the overall mean (Σ(Xi - X̄)²), while SSbetween involves calculating the squared deviations of group means from the overall mean, multiplied by the number of observations in each group (n*(X̄i - X̄)²).
After obtaining both SStotal and SSbetween, we can find SSwithin as the difference between them: SSwithin = SStotal - SSbetween. This value represents the variability within the groups and is essential for deriving the F-statistic. The F-statistic is calculated using the formula F = MSbetween / MSwithin, where MSbetween is the mean square for between-group differences computed by dividing SSbetween by its corresponding degrees of freedom (dfn), and MSwithin is determined similarly from SSwithin.
Once we have calculated the F-statistic, we can find the P-value associated with this statistic, which indicates the probability of observing such an F-statistic under the null hypothesis. If the P-value is less than the significance level (commonly set at 0.05), we reject the null hypothesis, concluding that there are significant differences between the group means.
To illustrate this with a practical example, let us say we conduct a study on teenage driving ages across various regions. Suppose we collect data indicating that the ages at which teenagers obtained their driving licenses differ markedly across five regions, with means of 16.5, 17.0, 17.5, 16.8, and 17.2 years. Utilizing Excel, we can feed this data into the ANOVA framework and perform the calculations outlined above to arrive at our F-statistic and P-value.
Furthermore, the implications of such findings can extend beyond academic discourse. In the realm of policy-making, understanding whether there are significant differences in the average age of obtaining a driver's license can inform legislative actions regarding drivers' education programs, and age restrictions, or even influence marketing strategies for driving schools targeting specific demographics.
In other scenarios, studies involving different demographic backgrounds, such as the frequency of eating out among various ethnic groups, can similarly utilize ANOVA to highlight whether significant disparities exist in behavior patterns, which could lead to tailored business approaches or community outreach initiatives.
To sum up, the analysis of variance is a vital statistical tool that facilitates the comparison of multiple group means, allowing researchers to make informed decisions regarding hypotheses related to population parameters. Its systematic approach to understanding variability among groups makes it indispensable in various research fields, from social sciences to health studies.
References
- Lane, D. M., & Steinberg, D. (2013). Introduction to Statistics. Online Statistics Education: A Multimedia Course of Study.
- Illowsky, B., & Dean, S. (2013). Introductory Statistics. OpenStax College.
- Rice, J. A. (2006). Mathematical Statistics and Data Analysis. Cengage Learning.
- McClave, J. T., & Sincich, T. (2017). Statistics. Pearson.
- Bluman, A. G. (2018). Elementary Statistics: A Step by Step Approach. McGraw-Hill Education.
- Movshon, J. A., & Ghosh, K. K. (2012). Statistics for Social Data Analysis. Sage Publications.
- Wackerly, D. D., Mendenhall, W., & Simmons, A. (2014). Mathematical Statistics With Applications. Cengage Learning.
- Moore, D. S., McCabe, G. P., & Craig, B. A. (2015). Introduction to the Practice of Statistics. W. H. Freeman.
- Weiss, N. A. (2016). Introductory Statistics. Pearson.
- Field, A. (2018). Discovering Statistics Using IBM SPSS Statistics. Sage Publications.