When studying an arbitrary population, that is not necessarily normally distributed, we can estimate the mean µ of the population by finding the sample mean Xbar from a large random sample of n measurements. But since Xbar is only an estimate, we must say that µ = Xbar +/- e, where e is some appropriate margin of error.
The size of the error depends on a desired level of confidence r, such as 0.90, 0.95, 0.98, or 0.99. The range of the estimate for µ, given by [Xbar - e, Xbar + e] is called a confidence interval. Each time a random sample is gathered, a different sample average Xbar will be found; thus, there will be a different confidence interval [Xbar - e, Xbar + e].
The level of confidence measures the probability that the unknown mean µ will actually be in a given confidence interval. So if the level of confidence is 0.95 and 100 random samples of size n are gathered, then around 95% of the resulting confidence intervals should contain µ. However, since µ is unknown, there is no real way of guaranteeing or even knowing if µ actually lies in a confidence interval. In any case, we could only be about 95% certain.
The level of confidence r and the sample size n determine a z-score z as follows: If Z ~ N(0,1) (the standard normal distribution), then z is the value such that P(-z <= Z <= z) = r.
The margin of error is determined by the sample deviation Sx, the z-score z, and the sample size n, and the population size. If we have a "large" population, then e is given by e = z*Sx / Sqrt(n). If we have a smaller population of size N, then we use the finite population correction factor so that e = z* Sx*Sqrt( (N -n) / (N - 1) ) / Sqrt(n).
Thus, we say
for large populations
for a smaller, finite population of size N
Usually, a sample of size n >= 30 is considered sufficient to use the analysis above. However in practice, if one wishes to obtain a reasonably small margin of error from a non-normal population, then much larger sample sizes will be needed.
The ZCONFINT program can be used to find such a confidence interval. To execute the program, first enter either 1 or 2 to specify a "large" population or a finite population. For Option 2, enter the population size N; otherwise just enter the sample size, the values of the sample mean and sample deviation, and the desired level of confidence (in decimal). The program displays the sample mean, the margin of error, and the confidence interval.
Example 1. A survey of 100 married couples was conducted to find out how many months they dated before getting married. The sample mean was Xbar = 11.41 with a sample deviation of Sx = 8.12. Find a 95% confidence interval for the true average number of months dated among all married couples if
(a) the survey was conducted in a metropolitan city.
(b) the survey was conducted in a church that has 480 married couples.
Solution. (a) After calling up the ZCONFINT program, first enter 1 for a large population. Next, enter 100 for SAMPLE SIZE, 11.41 for SAMPLE MEAN, 8.12 for SAMPLE DEV., and .95 for CONF. LEVEL. We find that µ ~ 11.41 +/- 1.5915, which gives us a 95% confidence interval of [9.8185, 13.0015].
(b) Reexecute the program, but first enter 2 for a finite population. Then enter the population size of 480 followed by the other summary statistics. Now we find that µ ~ 11.41 +/- 1.4175, which gives us a 95% confidence interval of [9.9925, 12.8275].
If we have a collection of measurements, then we can enter the data into the STAT Edit screen (LIST EDIT on the TI-86, APPS 6 on the TI-89) in order to compute the sample mean and sample deviation. We then can enter these values as variable symbols into the program.
Example 2. The measurements below are percentages of body fat from a random sample of men aged 20 to 29. Find a 90% confidence interval for the mean percentage body fat µ of all men aged 20 to 29.
Solution. First, enter the data into the STAT Edit screen (LIST EDIT on the TI-86, APPS 6 on the TI-89), and use the 1-Var Stats command (OneVar on the TI-86 and 89) to compute the statistics. We see that n = 36, Xbar = 14.4222 and Sx ~ 6.9497.
Next, call up the ZCONFINT program and enter 1 to assume we have a large population under consideration (all men age 20-29). We can type in the numerical values of n = 36, Xbar = 14.2222, and Sx = 6.9497 by hand, or we can access them symbolically within the program as outlined below:
On the TI-83: For SAMPLE SIZE, press VARS, press 5, press 1, press ENTER. For SAMPLE MEAN, press VARS, press 5, press 2, press ENTER. For SAMPLE DEV., press VARS, press 5, press 3, press ENTER.
On the TI-86: For SAMPLE SIZE, type 2nd ALPHA 9 to obtain n, press ENTER. For SAMPLE MEAN, press STAT (i.e., 2nd +), press F5, press F1 for Xbar, press ENTER. For SAMPLE DEV., press STAT (2nd +), press F5, press F3, press ENTER.
On the TI-89: For SAMPLE SIZE, type and enter nStat. For SAMPLE MEAN, press CHAR (i.e., 2nd +), press 2, then scroll down to Xbar (item A), press ENTER. For SAMPLE DEV., type and enter Sx.
We find that µ ~ 14.4222 +/- 1.9052 which gives [12.517, 16.1274] as a 90% confidence for the mean percentage body fat for men aged 20 - 29.
Consider the 36 measurements in Example 2 above. The true mean of these measurements is µ = 14.2222 and the true standard deviation is about 6.8525. Take a random sample of 20 of these measurements. (To do so, you can use the RANDOM program from the Miscellaneous section.)
Use the random sample to find a 95% confidence interval for µ . (When executing the ZCONFINT program, use the finite population size of 36 and use the true value of the standard deviation 6.8525 because it is known.)
Is µ in your interval?
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