6.2 Inference for the Mean in Practice
We have discussed how the sampling distribution of the sample mean follows a normal distribution when the population standard deviation, σ, is known and the t-distribution when it is not. In practice, we rarely know the population standard deviation. For larger samples, we can typically get away with using z according to the CLT. In summary, the majority of the time in which we opt to use t, we do not know σ and we have a small sample (n < 30).
Confidence Intervals for the Mean (σ Unknown)
The general format of a confidence interval is:
PE – MoE, PE + MoE
The population parameter is μ. The point estimate (PE) for μ is , the sample mean.
If the population standard deviation is not known, the margin of error for a population mean is:
MoE = () )
- is the t critical value with area to the right equal to
- use df = n – 1 degrees of freedom
- s = sample standard deviation
Example
The Federal Election Commission (FEC) collects information about campaign contributions and disbursements for candidates and political committees each election cycle. A political action committee (PAC) is a committee formed to raise money for candidates and campaigns. A Leadership PAC is a PAC formed by a federal politician (senator or representative) to raise money to help other candidates’ campaigns.[1]
The FEC has reported financial information for 556 Leadership PACs that operated during the 2011–2012 election cycle. The following table shows the total receipts during this cycle for a random selection of 30 Leadership PACs (in dollars).
| Receipt data | ||||
|---|---|---|---|---|
| $46,500.00 | $0 | $40,966.50 | $105,887.20 | $5,175.00 |
| $29,050.00 | $19,500.00 | $181,557.20 | $31,500.00 | $149,970.80 |
| $2,555,363.20 | $12,025.00 | $409,000.00 | $60,521.70 | $18,000.00 |
| $61,810.20 | $76,530.80 | $119,459.20 | $0 | $63,520.00 |
| $6,500.00 | $502,578.00 | $705,061.10 | $708,258.90 | $135,810.00 |
| $2,000.00 | $2,000.00 | $0 | $1,287,933.80 | $219,148.30 |
Figure 6.4: PAC receipt data
= $251,854.23
s = $521,130.41
Use this sample data to construct a 96% confidence interval for the mean amount of money raised by all Leadership PACs during the 2011–2012 election cycle. Use the Student’s t-distribution.
Note that we are not given the population standard deviation, only the standard deviation of the sample.
Solution
There are 30 measures in the sample, so n = 30, and df = 30 – 1 = 29.
CL = 0.96, so α = 1 – CL = 1 – 0.96 = 0.04.
α/2 = 0.02 tα/2 = t0.02 = 2.150
Margin of error = tα/2() = 2.150 (521130.41 ÷ √30) ∼ $204,561.66
– margin of error = 251,854.23 – 204,561.66 = $47,292.57
+ margin of error = 251,854.23 + 204,561.66 = $456,415.89
We estimate with 96% confidence that the mean amount of money raised by all Leadership PACs during the 2011–2012 election cycle lies between 47,292.57 and 456,415.89 dollars.
The 96% confidence interval is ($47,262, $456,447).
The difference between solutions arises from rounding differences.
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