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The following table shows the age profiles of all of the people in Ireland, aged between 15 and 65, who subscribed to an online streaming platform - Leaving Cert Mathematics - Question 7 - 2021
Question 7
The following table shows the age profiles of all of the people in Ireland, aged between 15 and 65, who subscribed to an online streaming platform. | Age Group | Pe... show full transcript
Worked Solution & Example Answer:The following table shows the age profiles of all of the people in Ireland, aged between 15 and 65, who subscribed to an online streaming platform - Leaving Cert Mathematics - Question 7 - 2021
Step 1
Draw a Histogram to represent the data.
Answer
To draw the histogram, create a bar graph where the x-axis represents the age groups (15-25, 25-35, 35-45, 45-55, and 55-65) and the y-axis represents the percentage of subscribers. Each bar's height corresponds to the percentage from the table provided:
- 15-25 years: 24%
- 25-35 years: 27%
- 35-45 years: 25%
- 45-55 years: 16%
- 55-65 years: 8%
Ensure the bars are evenly spaced and labeled correctly.
Step 2
In a survey of 1000 subscribers, aged between 15 and 65, how many people would you expect to be between 25 years and 55 years?
Answer
To find how many people are expected to be between 25 years and 55 years, sum the percentages of the respective age groups:
- 25-35: 27%
- 35-45: 25%
- 45-55: 16%
Total Percentage = 27 + 25 + 16 = 68%
Expected number of subscribers = 68% of 1000 = 0.68 * 1000 = 680 people.
Step 3
Use the table or the histogram to write down the age group which contains the median age.
Answer
To find the median age group, note that 50% of the data is below the median. From the cumulative percentages:
- 15-25 years: 24%
- 25-35 years: 51% Thus, the median falls into the 25-35 years age group.
Step 4
Use the mid-interval values of the groups in the table to estimate the mean age of the subscribers. Give your answer correct to one decimal place.
Answer
Calculate the mid-interval for each age group:
- 15-25: 20
- 25-35: 30
- 35-45: 40
- 45-55: 50
- 55-65: 60
Then use the formula for the mean:
ext{Mean Age} = rac{ ext{Sum of (mid-value * percentage)}}{100}
Substituting the mid-values and percentages:
ext{Mean Age} = rac{(20*24) + (30*27) + (40*25) + (50*16) + (60*8)}{100} = rac{(480 + 810 + 1000 + 800 + 480)}{100} = rac{3570}{100} = 35.7 Thus, the mean age is 35.7 years.
Step 5
Step 6
His survey revealed that 372 of the responders said they now have a Netflix account. Use your answer to Part (b)(i) above to create a 95% confidence interval for the percentage of the population that have a Netflix account.
Answer
To find a 95% confidence interval, use the formula:
where ( \tilde{p} = \frac{372}{540} = 0.6889 ), and ( z ) for 95% confidence is approximately 1.96.
Calculating the standard error (SE):
So the confidence interval is:
Expressing it as a percentage:
Step 7
Use your answer to Part (b)(ii) above to conduct the Hypothesis Test, at the 5% level of significance to test Aiden's claim that the percentage of subscribers to Netflix had changed.
Answer
State the hypotheses:
- Null Hypothesis (H0): The Netflix share of the market has not changed, or ( p = 0.65 )
- Alternative Hypothesis (H1): The Netflix share of the market has changed, or ( p ≠ 0.65 )
Based on the confidence interval obtained earlier, ( [64.6%, 73.2%] ), which does not include 65%. Therefore, we reject the null hypothesis at the 5% level of significance.
Conclusion: There is sufficient evidence to support Aiden’s claim that the percentage of subscribers to Netflix had changed.