# Confidence Intervals

## A Single Population Mean Using the Student's t-Distribution

In practice, we rarely know the population standard deviation. In the past, when the sample size was large, this unknown number did not present a problem to statisticians. They used the sample standard deviation s as an estimate for σ and proceeded as before to calculate a confidence interval with close-enough results. However, statisticians ran into problems when the sample size was small. A small sample size caused inaccuracies in the confidence interval. 
William S. Gosset (1876–1937) of the Guinness brewery in Dublin, Ireland, ran into this  problem. His experiments with hops and barley produced very few samples. Just replacing σ with s did not produce accurate results when he tried to calculate a confidence interval. He realized that he could not use a normal distribution for the calculation; he found that the actual distribution depends on the sample size. This problem led him to discover what is called the Student's . The name comes from the fact that Gosset wrote under the pen name Student.
Up until the mid-1970s, some statisticians used the normal distribution approximation for large sample sizes and used the Student's t-distribution only for sample sizes of at most 30. With graphing calculators and computers, the practice now is to use the Student's t-distribution whenever s is used as an estimate for σ.
If you draw a simple random sample of size n from a population that has an approximately normal distribution with mean μ and unknown population standard deviation σ and calculate the t-score t = 
 , then the t-scores follow a Student's . The t-score has the same interpretation as the : It measures how far 
  is from its mean μ. For each sample size n, there is a different Student's t-distribution.
The degrees of freedom (, n - 1, are the sample size minus 1.
2. The graph for the Student's t-distribution is similar to the standard normal curve.
3. The mean for the Student's t-distribution is zero, and the distribution is symmetric about zero.
4. The Student's t-distribution has more probability in its tails than the standard normal distribution.  shows the graphs of the student t-distribution for 1, 2 and 5 degrees of freedom: (v), compare to the standard normal distribution (in black).


5. The exact shape of the Student's t-distribution depends on the degrees of freedom. As the degrees of freedom increase, the graph of the Student's t-distribution becomes more like the graph of the standard normal distribution.
6. The underlying population of individual observations is assumed to be normally distributed with unknown population mean μ and unknown population standard deviation σ. The size of the underlying population is generally not relevant unless it is very small. If it is bell-shaped (normal), then the assumption is met and does not need discussion. Random sampling is assumed, but that is a completely separate assumption from normality.
Calculators and computers can easily calculate any Student's t-probabilities. The TI-83, 83+, and 84+ have a tcdf function to find the probability for given values of t. The grammar for the tcdf command is tcdf(lower bound, upper bound, degrees of freedom). However, for confidence intervals, we need to use inverse probability to find the value of t when we know the probability.
For the TI-84+, you can use the invT command on the DISTRibution menu. The invT command works similarly to the invnorm. The invT command requires two inputs: invT(area to the left, degrees of freedom). The output is the t-score that corresponds to the area we specified. 
 The TI-83 and 83+ do not have the invT command. (The TI-89 has an inverse T command.)
A probability table for the Student's t-distribution can also be used.  The table gives critical t-values that correspond to the confidence level (column) and degrees of freedom (row). (The TI-86 does not have an invT program or command, so if you are using that calculator, you need to use a probability table for the Student's t-distribution.) When using a t-table, note that some tables are formatted to show the confidence level in the column headings, while the column headings in some tables may show only corresponding area in one or both tails.

 A Student's t-table (see ) gives t-scores given the degrees of freedom and the right-tailed probability. The table is very limited. Calculators and computers can easily calculate any Student's  t-probabilities.
If the population standard deviation is  not known,  the error bound for a population mean is
1. 
 ,
2. 
  is the t-score with area to the right equal to 
 ,
3. use df = n – 1 degrees of freedom, and
4. s = sample standard deviation.
The format for the confidence interval is

### References

Bloomberg Businessweek. (n.d.). Retrieved from http://www.businessweek.com/
Environmental Working Group. (n.d.). Human toxome project: Mapping the pollution in people. Retrieved from http://www.ewg.org/sites/humantoxome/participants/participant-group.php?group=in+utero%2Fnewborn
Federal Election Commission. (n.d.). Disclosure data catalog: 2012 leadership PACs and sponsors. Retrieved from http://www.fec.gov/data/index.jsp
Federal Election Commission. (n.d.). Metadata description of leadership PAC list. Retrieved from http://www.fec.gov/finance/disclosure/metadata/metadataLeadershipPacList.shtml
Forbes. (2013). Americas best small companies. Retrieved from http://www.forbes.com/best-small-companies/list/
Forbes. (n.d.). Retrieved from http://www.forbes.com/
Microsoft Bookshelf. (n.d.).

### Chapter Review

In many cases, the researcher does not know the population standard deviation, σ, of the measure being studied. In these cases, it is common to use the sample standard deviation, s, as an estimate of σ. The normal distribution creates accurate confidence intervals when σ is known, but it is not as accurate when s is used as an estimate. In this case, the Student’s t-distribution is much better. Define a t-score using the following formula:
The t-score follows the Student’s t-distribution with n – 1 degrees of freedom. The confidence interval under this distribution is calculated with EBM = 
  where 
  is the t-score with area to the right equal to 
 , s is the sample standard deviation, and n is the sample size. Use a table, calculator, or computer to find 
  for a given α.

### Formula Review

s = the standard deviation of sample values


  is the formula for the t-score, which measures how far away a measure is from the population mean in the Student’s t-distribution.
df = n – 1; the degrees of freedom for a Student’s t-distribution, where n represents the size of the sample
T~t the random variable, T, has a Student’s t-distribution with df degrees of freedom

  = the error bound for the population mean when the population standard deviation is unknown

  is the t-score in the Student’s t-distribution with area to the right equal to 
 
The general form for a confidence interval for a single mean, population standard deviation unknown, Student's t is given by  (lower bound, upper bound)
= (point estimate – EBM, point estimate + EBM)
Use the following information to answer the next five exercises: A hospital is trying to cut down on emergency room wait times. It is interested in the amount of time patients must wait before being called back to be examined. An investigation committee randomly surveyed 70 patients. The sample mean was 1.5 hr, with a sample standard deviation of 0.5 hr.
Use the following information to answer the next six exercises: One hundred eight Americans were surveyed to determine the number of hours they spend watching television each month. It was revealed that they watch an average of 151 hours each month, with a standard deviation of 32 hours. Assume that the underlying population distribution is normal.
Use the following information to answer the next 13 exercises: The data in  are the result of a random survey of 39 national flags (with replacement between picks) from various countries. We are interested in finding a confidence interval for the true mean number of colors on a national flag. Let X = the number of colors on a national flag.
Construct a 95 percent confidence interval for the true mean number of colors on national flags.

### Homework

Use the following information to answer the next two exercises: A quality control specialist for a restaurant chain takes a random sample of size 12 to check the amount of soda served in the 16-oz serving size. The sample mean is 13.30, with a sample standard deviation of 1.55. Assume the underlying population is normally distributed.