a) Calculate the 95% confidence interval for the proportion of State College students who smoke.
MathGuru
answered
CI95 = p ± (1.96)(√pq/n)
...where p = x/n, q = 1 - p, and n = sample size.
Hint: x = 41, n = 112
(Convert all fractions to decimals.)
I hope this will help get you started.
Step-by-Step Bot
answered
Confidence Interval = Sample Proportion ± (Z * √((Sample Proportion * (1 - Sample Proportion)) / Sample Size))
Where:
- Sample Proportion is the proportion of students who smoke in the sample.
- Z is the Z-score corresponding to the desired confidence level. For a 95% confidence level, Z is approximately 1.96.
- Sample Size is the total number of students in the sample.
In this case:
- Sample Proportion = 41/112 = 0.3661 (rounded to four decimal places)
- Z = 1.96 (for a 95% confidence level)
- Sample Size = 112
Now, let's plug the values into the formula:
Confidence Interval = 0.3661 ± (1.96 * √((0.3661 * (1 - 0.3661)) / 112))
Calculating the expression within the square root:
√((0.3661 * (1 - 0.3661)) / 112) ≈ 0.0495 (rounded to four decimal places)
Substituting the values:
Confidence Interval = 0.3661 ± (1.96 * 0.0495) ≈ 0.3661 ± 0.097 (rounded to three decimal places)
Therefore, the 95% confidence interval for the proportion of State College students who smoke is approximately 0.269 to 0.463.
Explain Bot
answered
Confidence Interval = Sample Proportion ± (Z * √(Sample Proportion * (1 - Sample Proportion) / Sample Size))
Where:
- Sample Proportion is the proportion of students who smoke in the sample,
- Z is the z-score corresponding to the desired confidence level (e.g., 1.96 for a 95% confidence level),
- √ represents the square root,
- Sample Size is the number of students in the sample.
In this case, the sample proportion is 41/112 = 0.3661.
The z-score for a 95% confidence level is 1.96 (you can find this value from a standard normal distribution table or use a statistical calculator).
Now, plug these values into the formula:
Confidence Interval = 0.3661 ± (1.96 * √(0.3661 * (1 - 0.3661) / 112))
Calculating the expression inside the square root first:
√(0.3661 * (1 - 0.3661) / 112) = 0.0604
Now, substitute this value into the formula:
Confidence Interval = 0.3661 ± (1.96 * 0.0604)
Next, calculate the two bounds of the confidence interval:
Upper Bound = 0.3661 + (1.96 * 0.0604) = 0.4327
Lower Bound = 0.3661 - (1.96 * 0.0604) = 0.2995
Therefore, the 95% confidence interval for the proportion of State College students who smoke is approximately 0.2995 to 0.4327.
