a. In a random sample of 500 observations , we found the proportion of successes to be 48%. Estimate with 95% confidence the population proportion of successes.
B. Repeat part (a) with N=200
C. Repeat part (a) with N=1000
I need to know how to do this in excel?
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To calculate the confidence interval for a proportion in Excel, you can use the CONFIDENCE.T function. The formula for the confidence interval is given by:
CONFIDENCE.T(alpha, standard_deviation, sample_size)
where:
- alpha: the significance level (e.g., 0.05 for a 95% confidence interval)
- standard_deviation: the standard deviation of the sample proportion
- sample_size: the sample size
Now, let's calculate the confidence intervals for parts (a), (b), and (c) using Excel:
a. Random sample of 500 observations, with a proportion of successes of 48%:
- Sample proportion of successes (p-hat) = 48% = 0.48
- Sample size (n) = 500
1. First, calculate the standard deviation of the sample proportion:
- Standard deviation (s) = sqrt(p-hat * (1 - p-hat) / n)
In Excel, use the formula: "=SQRT(0.48*(1-0.48)/500)"
2. Next, calculate the confidence interval:
- Confidence interval = p-hat ± (z * s)
In Excel, use the formula: "=0.48-(NORM.S.INV(1-0.05/2)*(SQRT(0.48*(1-0.48)/500)))" for the lower limit and "=0.48+(NORM.S.INV(1-0.05/2)*(SQRT(0.48*(1-0.48)/500)))" for the upper limit. This assumes a 95% confidence level.
b. Sample size of 200:
- Repeat the above steps using a sample size of 200 instead of 500.
c. Sample size of 1000:
- Repeat the above steps using a sample size of 1000 instead of 500.
By applying the formulas correctly, Excel will calculate the confidence intervals for you. Remember to replace the values in the formulas with the corresponding sample size and proportion of successes for each case.