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In Excel, confidence intervals can be calculated using the CONFIDENCE function which takes in three arguments: alpha, standard deviation, and size. Alpha is the level of confidence, standard deviation is the standard deviation of the population, and size is the sample size. The function returns the confidence interval of the mean. Additionally, formulas such as TINV can be used to calculate the confidence interval given the alpha level and degrees of freedom.
A confidence interval is a range of values that is likely to contain a population parameter with a certain level of confidence. It is calculated using the following general formula:
Confidence Interval = (point estimate) +/- (critical value)*(standard error)
This formula creates an interval with a lower bound and an upper bound, which likely contains a population parameter with a certain level of confidence:
Confidence Interval = [lower bound, upper bound]
This tutorial explains how to calculate the following confidence intervals in Excel:
1. Confidence Interval for a Mean
2. Confidence Interval for a Difference in Means
3. Confidence Interval for a Proportion
4. Confidence Interval for a Difference in Proportions
Let’s jump in!
Example 1: Confidence Interval for a Mean
We use the following formula to calculate a confidence interval for a mean:
Confidence Interval = x +/- z*(s/√n)
where:
- x: sample mean
- z: the chosen z-value
- s: sample standard deviation
- n: sample size
Example: Suppose we collect a random sample of turtles with the following information:
- Sample size n = 25
- Sample mean weight x = 300
- Sample standard deviation s = 18.5
The following screenshot shows how to calculate a 95% confidence interval for the true population mean weight of turtles:
The 95% confidence interval for the true population mean weight of turtles is [292.75, 307.25].
Example 2: Confidence Interval for a Difference in Means
We use the following formula to calculate a confidence interval for a difference in population means:
Confidence interval = (x1–x2) +/- t*√((sp2/n1) + (sp2/n2))
where:
- x1, x2: sample 1 mean, sample 2 mean
- t: the t-critical value based on the confidence level and (n1+n2-2) degrees of freedom
- sp2: pooled variance, calculated as ((n1-1)s12 + (n2-1)s22) / (n1+n2-2)
- t: the t-critical value
- n1, n2: sample 1 size, sample 2 size
Example: Suppose we want to estimate the difference in mean weight between two different species of turtles, so we go out and gather a random sample of 15 turtles from each population. Here is the summary data for each sample:
Sample 1:
- x1 = 310
- s1 = 18.5
- n1 = 15
Sample 2:
- x2 = 300
- s2 = 16.4
- n2 = 15
The following screenshot shows how to calculate a 95% confidence interval for the true difference in population means:
The 95% confidence interval for the true difference in population means is [-3.08, 23.08].
Example 3: Confidence Interval for a Proportion
We use the following formula to calculate a confidence interval for a proportion:
Confidence Interval = p +/- z*√p(1-p) / n
where:
- p: sample proportion
- z: the chosen z-value
- n: sample size
Example: Suppose we want to estimate the proportion of residents in a county that are in favor of a certain law. We select a random sample of 100 residents and ask them about their stance on the law. Here are the results:
- Sample size n = 100
- Proportion in favor of law p = 0.56
The following screenshot shows how to calculate a 95% confidence interval for the true proportion of residents in the entire county who are in favor of the law:
The 95% confidence interval for the true proportion of residents in the entire county who are in favor of the law is [.463, .657].
Example 4: Confidence Interval for a Difference in Proportions
We use the following formula to calculate a confidence interval for a difference in proportions:
Confidence interval = (p1–p2) +/- z*√(p1(1-p1)/n1 + p2(1-p2)/n2)
where:
- p1, p2: sample 1 proportion, sample 2 proportion
- z: the z-critical value based on the confidence level
- n1, n2: sample 1 size, sample 2 size
Example: Suppose we want to estimate the difference in the proportion of residents who support a certain law in county A compared to the proportion who support the law in county B. Here is the summary data for each sample:
Sample 1:
- n1 = 100
- p1 = 0.62 (i.e. 62 out of 100 residents support the law)
Sample 2:
- n2 = 100
- p2 = 0.46 (i.e. 46 our of 100 residents support the law)
The following screenshot shows how to calculate a 95% confidence interval for the true difference in proportion of residents who support the law between the counties:
The 95% confidence interval for the true difference in proportion of residents who support the law between the counties is [.024, .296].
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