1.Given the following samples of three groups; group1; 85,86,88,75,78,94,98,79,71,80, group2; 91,92,93,85,87,84,82,88,95,96 and group3; 79,78,88,94,92,85,83,85,82,81.
a. You have been asked to compare the means of group 1 and group 2, given that the two groups are independent, conduct the test for Equal variance.
b. suppose group 3 are sample scores for statistics mid semester exams for DCM class. Calculate the probability that the student scored above 90%, given that the population standard deviation for the scores was 4% and the population mean was 85%.
c. Using information from part c, calculate the probability of scoring less than or equal to each score in group 3.
d. State all the assumptions that have to be met when conducting hypotheses tests in all the tests covered in this class.
a. To compare the means of group 1 and group 2, we can conduct a two-sample t-test assuming equal variances.
To test for equal variances, we can use the F-test. The null hypothesis for testing equal variances is that the variances of both groups are equal, while the alternative hypothesis is that the variances are not equal.
Using the F-test:
Step 1: Calculate the variances of both groups.
- Group 1: Variance1 = [(85-82.545)^2 + (86-82.545)^2 + ... + (80-82.545)^2]/(n1-1)
- Group 2: Variance2 = [(91-88.2)^2 + (92-88.2)^2 + ... + (96-88.2)^2]/(n2-1)
Step 2: Calculate the F-statistic
- F-statistic = Variance1 / Variance2
Step 3: Compare the F-statistic to the critical value from the F-distribution with (n1-1) and (n2-1) degrees of freedom at the desired significance level. If the F-statistic is greater than the critical value, we reject the null hypothesis and conclude that the variances are not equal. If the F-statistic is not greater than the critical value, we fail to reject the null hypothesis and conclude that the variances are equal.
b. To calculate the probability that a student scored above 90% for the statistics mid-semester exam, we can use the population mean and standard deviation.
Assuming the scores are normally distributed, we can use the Z-score formula:
Z = (X - μ) / σ
Where:
X = 90% (desired score)
μ = 85% (population mean)
σ = 4% (population standard deviation)
Using the Z-table or a Z-score calculator, we can find the probability of scoring above 90%.
c. To calculate the probability of scoring less than or equal to each score in group 3, we can convert the distribution to a standard normal distribution using the Z-score formula:
Z = (X - μ) / σ
where:
X = individual score in group 3
μ = 85% (population mean)
σ = 4% (population standard deviation)
With the Z-scores calculated, we can use the Z-table to find the probability of scoring less than or equal to each score.
d. The assumptions that have to be met when conducting hypothesis tests are:
1. Random Sampling: The data should be collected through a random sampling process to ensure that it is a representative sample of the population.
2. Independence: The observations should be independent of each other, meaning that one observation does not influence another.
3. Normality: The data should be approximately normally distributed. This assumption is more important for smaller sample sizes. If the sample size is large enough (usually n > 30), the Central Limit Theorem allows for approximation to normality even if the underlying population is not exactly normal.
4. Homogeneity of Variance: The variances of the groups being compared should be equal, especially when conducting hypothesis tests for the means of two or more groups using t-tests or ANOVA.
It is important to check and meet these assumptions before conducting any hypothesis tests to ensure the validity of the results.
a. To compare the means of group 1 and group 2 and test for equal variance, you need to perform an independent samples t-test:
Step 1: Calculate the mean and standard deviation for each group.
For group 1: mean1 = (85+86+88+75+78+94+98+79+71+80)/10
For group 2: mean2 = (91+92+93+85+87+84+82+88+95+96)/10
For group 1: standard_deviation1 = sqrt(sum((x-mean1)^2)/n)
For group 2: standard_deviation2 = sqrt(sum((x-mean2)^2)/n)
Step 2: Calculate the test statistic t.
t = (mean1 - mean2) / sqrt((standard_deviation1^2/n1) + (standard_deviation2^2/n2))
Step 3: Find the degrees of freedom (df).
df = n1 + n2 - 2
Step 4: Determine the critical value for the desired significance level and degrees of freedom. Compare the calculated t-value with the critical value.
Step 5: Make a decision based on the comparison of the calculated t-value and critical value.
b. To calculate the probability that a student scored above 90% on the statistics mid-semester exam, we can use the Z-score formula since we know the population mean and standard deviation:
Step 1: Calculate the Z-score.
Z = (90 - 85) / 4
Step 2: Look up the Z-score in a standard normal distribution table to find the corresponding probability.
c. To calculate the probability of scoring less than or equal to each score in group 3, we can again use the Z-score formula and the information provided in part b:
Step 1: Calculate the Z-score for each score in group 3.
Z = (score - mean) / standard_deviation
Step 2: Look up the Z-score for each score in a standard normal distribution table to find the corresponding probability.
d. The assumptions that need to be met when conducting hypothesis tests in this class include:
1. Random sample: The data under investigation should be obtained through a random sampling process to ensure representativeness.
2. Independence: The observations within the sample should be independent of each other.
3. Normality: The population should follow a normal distribution, particularly for smaller sample sizes. If the sample size is large enough, this assumption can be relaxed due to the central limit theorem.
4. Equal variance: The variances of the populations being compared should be equal. This assumption is not required for some tests, such as Welch's t-test if the sample sizes and variances are unequal.
5. Homoscedasticity: The variances of the residuals should be constant across all levels of the independent variables.
It is important to note that not all tests require the same assumptions, and the specific test being used should be considered to ensure the appropriate assumptions are met.