Answer:
It seems like you want to compare the average cost of movie tickets in London and New York City using sample data and the given standard deviations. To do this, you can perform a hypothesis test. In this case, you might want to use a two-sample t-test since you have two independent samples (London and NYC) and want to compare their means.
Here are the steps to perform the t-test:
**Step 1: Define the null and alternative hypotheses:**
Null Hypothesis (H0): The average cost of movie tickets in London is equal to the average cost of movie tickets in New York City.
Alternative Hypothesis (Ha): The average cost of movie tickets in London is not equal to the average cost of movie tickets in New York City.
Mathematically, this can be represented as:
H0: μL = μNYC
Ha: μL ≠ μNYC
Where:
- μL represents the population mean of movie ticket prices in London.
- μNYC represents the population mean of movie ticket prices in New York City.
**Step 2: Set the significance level (alpha):**
You need to decide on a significance level (alpha), which is typically set at 0.05 or 5%.
**Step 3: Collect and summarize the data:**
You have the following information:
- For London: Sample mean (xL) = $19.63, Sample standard deviation (σL) = $3.20, Sample size (nL) = 25
- For New York City: Sample mean (xNYC) = $10.25, Sample standard deviation (σNYC) = $2.57, Sample size (nNYC) = 25
**Step 4: Perform the t-test:**
You can use a formula for a two-sample t-test with unequal variances since the standard deviations are different:
\[t = \frac{{xL - xNYC}}{{\sqrt{\frac{{σL^2}}{{nL}} + \frac{{σNYC^2}}{{nNYC}}}}}\]
Substitute the values:
\[t = \frac{{19.63 - 10.25}}{{\sqrt{\frac{{3.20^2}}{{25}} + \frac{{2.57^2}}{{25}}}}\]
**Step 5: Calculate the degrees of freedom:**
You'll need to calculate the degrees of freedom using the formula:
\[df = \frac{{\left(\frac{{σL^2}}{{nL}} + \frac{{σNYC^2}}{{nNYC}}\right)^2}}{{\frac{{\left(\frac{{σL^2}}{{nL}}\right)^2}}{{nL - 1}} + \frac{{\left(\frac{{σNYC^2}}{{nNYC}}\right)^2}}{{nNYC - 1}}}}\]
Substitute the values:
\[df = \frac{{\left(\frac{{3.20^2}}{{25}} + \frac{{2.57^2}}{{25}}\right)^2}}{{\frac{{\left(\frac{{3.20^2}}{{25}}\right)^2}}{{25 - 1}} + \frac{{\left(\frac{{2.57^2}}{{25}}\right)^2}}{{25 - 1}}}}\]
**Step 6: Find the critical t-value:**
Using the t-distribution table or a calculator, find the critical t-value for a two-tailed test with your chosen alpha level (e.g., 0.05) and degrees of freedom (df) from Step 5.
**Step 7: Calculate the t-statistic:**
Substitute the values into the formula:
\[t = \frac{{19.63 - 10.25}}{{\sqrt{\frac{{3.20^2}}{{25}} + \frac{{2.57^2}}{{25}}}}\]
**Step 8: Compare the t-statistic with the critical t-value:**
If the absolute value of the calculated t-statistic from Step 7 is greater than the critical t-value, you can reject the null hypothesis (H0). Otherwise, you fail to reject it.
**Step 9: Make a conclusion:**
Compare the t-statistic with the critical t-value and consider the significance level. If the t-statistic falls in the rejection region, you can conclude that there is a significant difference in the average movie ticket prices between London and New York City. Otherwise, you cannot conclude a significant difference.
Explanation: