7.2 Inference for the Mean in Practice
We have discussed 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 we opt to use t is when 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:
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 (MoE) for a population mean is:
- ,
- is the t critical value with area to the right equal to ,
- use df = n – 1 degrees of freedom, and
- 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)
| $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 |
= $251,854.23
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.
Your turn!
A random sample of statistics students were asked to estimate the total number of hours they spend watching television in an average week. The responses are recorded in the figure below. Use this sample data to construct a 98% confidence interval for the mean number of hours statistics students will spend watching television in one week.
| 0 | 3 | 1 | 20 | 9 |
| 5 | 10 | 1 | 10 | 4 |
| 14 | 2 | 4 | 4 | 5 |
Hypothesis Tests for the Mean (σ Unknown)
Remember, we will use the t-distribution when the population standard deviation is unknown and the distribution of the sample mean is approximately normal.
If you are testing a single population mean, and we decide to use t, the steps say the same, but our test statistic will change s