1.
The following sample data shows the use of plastic (debit and credit) cards versus
cash/cheque by four age groups:
Age Group 18-24, 25-34, 35-44, 45 and over
Payment Method
plastic 21, 27, 27, 36
cash or cheque 21, 36, 42, 90
Test for independence between method of payment and age group. Use the p-value approach
and α = 0.05. To make it easier to compare answers, please round off to two decimal places.
2. Consider the following effects of three treatments
A B C
23 33 25
34 25 23
26 35 32
33 29 35
24 32 22
24 26 34
22 26 29
31 33 36
32 33
31 35
mean 27.1 30.2 30.4
Sample Standard deviation 4.8 3.5 5.3
At α = 0.05 is there a significant difference between the treament means? To make our
numbers easier to compare, you can round off the overall mean to one decimal place. Don’t
forget to provide an ANOVA table
can anyone Please help, really struggling
Sure, I can help you. Let's tackle each question step by step.
Question 1:
To test for independence between method of payment and age group, we need to perform a chi-square test. Specifically, we will perform a chi-square test for independence. Here's how you can do it:
Step 1: Set up the hypotheses:
- Null hypothesis (H0): Method of payment and age group are independent.
- Alternative hypothesis (H1): Method of payment and age group are dependent.
Step 2: Calculate the expected frequencies:
To determine the expected frequencies, we assume that the null hypothesis is true and calculate the expected frequencies using the formula: (row total x column total) / grand total.
Age Group plastic cash or cheque Total
18-24 (21+21) (27+36) 87
25-34 (27+21) (27+36) 111
35-44 (27+21) (42+36) 126
45 and over (36+21) (90+42) 189
Total 105 195 513
Step 3: Calculate the chi-square test statistic:
The chi-square test statistic is given by the formula: X² = Σ (O - E)² / E, where O is the observed frequency and E is the expected frequency.
Using the values from your sample data, calculate the chi-square test statistic.
Step 4: Determine the degrees of freedom:
The degrees of freedom for a chi-square test of independence is calculated as (number of rows - 1) x (number of columns - 1). In this case, we have (4-1) x (2-1) = 3.
Step 5: Determine the p-value:
Using the chi-square distribution table or a statistical software, find the p-value associated with the calculated chi-square test statistic and the degrees of freedom.
Step 6: Make a decision:
Compare the p-value to the significance level (α = 0.05). If the p-value is less than α, we reject the null hypothesis and conclude that there is evidence of dependence between method of payment and age group. Otherwise, if the p-value is greater than or equal to α, we fail to reject the null hypothesis.
Question 2:
To determine if there is a significant difference between the treatment means, we will use a One-Way ANOVA (Analysis of Variance) test. Here's how to approach it:
Step 1: Set up the hypotheses:
- Null hypothesis (H0): The means of the treatments are equal.
- Alternative hypothesis (H1): At least one of the treatment means is different.
Step 2: Calculate the Sum of Squares (SS):
Calculate the Sum of Squares (SS) for each factor: Treatment, Error, and Total. Then, calculate the Mean Square (MS) by dividing each Sum of Squares by its respective degrees of freedom. Finally, calculate the F statistic using the formula: F = MS_Treatment / MS_Error.
Step 3: Determine the p-value:
Using the F-distribution table or statistical software, find the p-value associated with the calculated F statistic and the degrees of freedom.
Step 4: Make a decision:
Compare the p-value to the significance level (α = 0.05). If the p-value is less than α, we reject the null hypothesis and conclude that there is a significant difference between the treatment means. Otherwise, if the p-value is greater than or equal to α, we fail to reject the null hypothesis.
To make it easier to compare answers, round off the means to one decimal place.
Additionally, provide an ANOVA table summarizing the results, including the sources of variation, degrees of freedom, sum of squares, mean squares, F statistic, and the p-value.
I hope this helps you in conducting the statistical tests. If you need further assistance or clarifications, feel free to ask!