Joan moves into her apartment and wants to purchase a new couch. She wants to determine if there is any difference between the average cost of couches at two different stores. Test the hypothesis that there is no difference at a=0.05
Store 1 Store 2
x(sample mean).........$650... $730
o(sample deviation).... $61... $78
n....................... 24.... 21
Ho: u1-u2 = 0
Ha: u1-u2 is not 0
test-statistic = -3.8563
p-value = 0.00038
df = 43
Conclusion:Since p-value is less than alpha=5%, Reject Ho.
There is at difference in the average cost of couches at these two stores.
To test the hypothesis that there is no difference between the average cost of couches at Store 1 and Store 2, we can perform a two-sample t-test.
1. State the null and alternative hypotheses:
- Null hypothesis (H₀): There is no difference between the average cost of couches at Store 1 and Store 2.
- Alternative hypothesis (H₁): There is a difference between the average cost of couches at Store 1 and Store 2.
2. Determine the significance level (α) and set the critical value(s):
The significance level (α) is given as 0.05.
3. Calculate the test statistic:
- Calculate the pooled standard deviation (Sp):
Sp = sqrt(((n1 - 1) * o1^2 + (n2 - 1) * o2^2) / (n1 + n2 - 2))
where o1 and o2 are the sample deviations for Store 1 and Store 2, respectively.
- Calculate the t-test statistic:
t = (x₁ - x₂) / (Sp * sqrt((1 / n1) + (1 / n2)))
where x₁ and x₂ are the sample means for Store 1 and Store 2, respectively.
4. Determine the critical region:
- Find the degrees of freedom (df):
df = n1 + n2 - 2
- Look up the critical value(s) from the t-distribution table using the degrees of freedom and significance level (α).
5. Compare the test statistic to the critical value(s):
- If the absolute value of the test statistic is greater than the critical value(s), reject the null hypothesis.
- If the absolute value of the test statistic is less than or equal to the critical value(s), fail to reject the null hypothesis.
6. Draw a conclusion:
- If the null hypothesis is rejected, there is evidence to suggest a difference between the average cost of couches at Store 1 and Store 2.
- If the null hypothesis is not rejected, there is insufficient evidence to suggest a difference between the average cost of couches at Store 1 and Store 2.
Please plug in the given values into the formulas to calculate the test statistic and compare it with the critical value.
To test the hypothesis that there is no difference in the average cost of couches at two different stores, we can use a two-sample t-test. This test compares the means of two independent samples and determines if the difference is statistically significant.
Here's how you can conduct the two-sample t-test:
Step 1: State the null and alternative hypotheses:
The null hypothesis (H0): There is no difference in the average cost of couches at Store 1 and Store 2.
The alternative hypothesis (Ha): There is a difference in the average cost of couches at Store 1 and Store 2.
Step 2: Set the significance level (alpha):
In this case, the significance level is given as α=0.05.
Step 3: Calculate the test statistic:
The formula for the two-sample t-test is:
t = (x1 - x2) / √[(s1²/n1) + (s2²/n2)]
where x1 and x2 are the sample means, s1 and s2 are the sample standard deviations, and n1 and n2 are the sample sizes.
Given the data:
Sample mean (x1) = $650
Sample mean (x2) = $730
Sample deviation (s1) = $61
Sample deviation (s2) = $78
Sample size (n1) = 24
Sample size (n2) = 21
Using these values, we can calculate the test statistic.
Step 4: Determine the critical value:
For a two-sample t-test with a 0.05 significance level and unequal sample sizes, degrees of freedom are calculated using the Welch-Satterthwaite equation: df = (s1²/n1 + s2²/n2)² / [(s1²/n1)² / (n1-1) + (s2²/n2)² / (n2-1)].
Using this equation with the provided sample data, we can find the degrees of freedom.
Step 5: Compute the p-value:
Using the test statistic and degrees of freedom calculated in the previous steps, we can find the p-value. The p-value is the probability of obtaining a test statistic as extreme or more extreme than the observed value, assuming the null hypothesis is true.
Step 6: Compare the p-value with the significance level:
If the p-value is less than the significance level (α), then we reject the null hypothesis. If the p-value is greater than α, we fail to reject the null hypothesis.
After following these steps, you will have completed the hypothesis test and determined if there is a statistically significant difference in the average cost of couches at Store 1 and Store 2.