Australia now enter the furthest stage of the tournament in the football team's history and will go into the game against England in the semi-finals of the FIFA Women's World Cup 2023. A survey of 1000 randomly selected Australian football fans are asked if Australia can beat England in the semi-finals. 76% of them believe that Australia can beat England in the semi-finals. Construct a 95% confidence interval around the proportion of Australian football fans who believe that Australia can beat England in the semi-finals. What is the lower bound of this interval (round your answer to four decimal places) ?
Question
Australia now enter the furthest stage of the tournament in the football team's history and will go into the game against England in the semi-finals of the FIFA Women's World Cup 2023. A survey of 1000 randomly selected Australian football fans are asked if Australia can beat England in the semi-finals. 76% of them believe that Australia can beat England in the semi-finals. Construct a 95% confidence interval around the proportion of Australian football fans who believe that Australia can beat England in the semi-finals. What is the lower bound of this interval (round your answer to four decimal places) ?
Solution
To construct a 95% confidence interval for the proportion, we can use the formula:
CI = p̂ ± Z * sqrt[(p̂(1 - p̂) / n]
where:
- p̂ is the sample proportion
- Z is the Z-score for the desired confidence level
- n is the sample size
In this case:
- p̂ = 0.76 (the sample proportion)
- Z = 1.96 (the Z-score for a 95% confidence level)
- n = 1000 (the sample size)
Substituting these values into the formula, we get:
CI = 0.76 ± 1.96 * sqrt[(0.76 * 0.24) / 1000]
To find the lower bound of the confidence interval, we subtract the margin of error from the sample proportion:
Lower bound = 0.76 - 1.96 * sqrt[(0.76 * 0.24) / 1000]
Calculating the above expression gives a lower bound of approximately 0.7328 when rounded to four decimal places.
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