An online site presented this question, 'Would the recent norovirus outbreak deter you from taking a cruise?' Among the 34 comma 203 people who responded, 66% answered 'yes'. Use the sample data to construct a 90% confidence interval estimate for the proportion of the population of all people who would respond 'yes' to that question
To construct a confidence interval for the proportion of the population that would respond 'yes' to the question, we can use the following formula:
CI = p̂ ± z * √(p̂(1-p̂)/n)
Where:
CI = Confidence Interval
p̂ = Sample proportion
z = Z-score (corresponding to the desired confidence level)
n = Sample size
In this case, the sample proportion (p̂) is 66%, which can be expressed as 0.66. The sample size (n) is 34,203.
First, let's find the Z-score for a 90% confidence level. The Z-score values are commonly found in Z-tables or can be calculated using statistical software. For a 90% confidence level, the Z-score is approximately 1.645.
Now, let's calculate the confidence interval:
CI = 0.66 ± 1.645 * √(0.66 * (1-0.66) / 34,203)
= 0.66 ± 1.645 * √(0.66 * 0.34 / 34,203)
= 0.66 ± 1.645 * √(0.2244 / 34,203)
= 0.66 ± 1.645 * √(0.0000065601)
= 0.66 ± 1.645 * 0.0025618
= 0.66 ± 0.0042146
Therefore, the 90% confidence interval estimate for the proportion of the population of all people who would respond 'yes' to the question is (0.6558, 0.6642).
To construct a confidence interval estimate for the proportion of the population who would respond 'yes' to the question, we can use the sample data provided.
Given:
Sample proportion (p̂) = 66% = 0.66
Sample size (n) = 34,203
Confidence level (α) = 90% (which corresponds to a significance level of 0.1 or 10%)
To calculate the confidence interval estimate, we can use the formula:
Confidence Interval = p̂ ± Z * √(p̂(1-p̂)/n)
Where:
Z is the z-score that corresponds to the desired confidence level
To find the Z-score for the 90% confidence level, we can use a standard normal distribution table or a calculator, which gives us a value of approximately 1.645.
Plugging in the values into the formula, we get:
Confidence Interval = 0.66 ± 1.645 * √((0.66*(1-0.66))/34,203)
Calculating the values in the formula:
Confidence Interval = 0.66 ± 1.645 * √(0.22704/34,203)
Confidence Interval = 0.66 ± 1.645 * 0.00455
Confidence Interval = 0.66 ± 0.00749
Finally, the confidence interval estimate for the proportion of the population who would respond 'yes' to the question is:
0.66 ± 0.00749
This means we are 90% confident that the true proportion of the population who would respond 'yes' lies between 0.65251 (0.66 - 0.00749) and 0.66749 (0.66 + 0.00749).