Practice problems for confidence interval for sigma_x, the standard deviation for the population

We have learned the methods for finding confidence intervals for proportions and averages of populations given similar statistics from samples. There is also a method for estimating the standard deviation of a population and giving a confidence level to that interval.

Let's say we took a sample of 28 scores and got a standard deviation of sx = 16.689, rounded to three places after the decimal. The degrees of freedom is n-1, which in this case is 27. Let's look at the Chi square table at the line that corresponds to d.f. = 27.

____0.995__0.99___0.975__0.95___0.90___||_0.10___0.05___0.025__0.01___0.005
27__11.808 12.879 14.573 16.151 18.114 || 36.741 40.113 43.194 46.963 49.645

The denominators in the formulas shown above are taken from the following columns.

90% confidence: Chi square Big comes from the 0.05 column, Chi square Small comes from the 0.95 column.

95% confidence: Chi square Big comes from the 0.025 column, Chi square Small comes from the 0.975 column.

99% confidence: Chi square Big comes from the 0.005 column, Chi square Small comes from the 0.995 column.

In this example, the formulas would look as follows.

90% confidence interval: sqrt(16.689^2*27/40.113) < sigmax < sqrt(16.689^2*27/16.151)

95% confidence interval: sqrt(16.689^2*27/43.194) < sigmax < sqrt(16.689^2*27/14.573)

99% confidence interval: sqrt(16.689^2*27/49.645) < sigmax < sqrt(16.689^2*27/11.808)

If n-1 is not one of the values in the degrees of freedom chart, use the next lowest number on the list.

Exercise #1: Find the values from the equations listed above, rounded to the nearest thousandth.

Exercise #2: Find the confidence intervals for 90%, 95% and 99% if n = 102 and sigmax = 0.62. Round the answers to two places after the decimal.

Answers in the comments.

Practice for matched pairs.

Was the price of silver in 2007 significantly different than it was in 2008?

Side by side, we have two lists of prices of silver, the highest price in a given month in 2007, followed by the highest price in that same month in 2008. Take the differences in the prices and find the average and standard deviation. The size of the list is 12, so the degrees of freedom are 11. If we assume we did not know which year showed higher prices when we started this experiment, it make sense to make this a two-tailed test. Just for a change of pace, let us use the 90% confidence level.

Mo.___2007___2008
Jan.__13.45__16.23
Feb.__14.49__19.81
Mar.__13.34__20.67
Apr.__14.01__17.74
May___12.90__18.19
Jun.__13.19__17.50
Jul.__12.86__18.84
Aug.__12.02__15.27
Sep.__12.77__12.62
Oct.__14.17__11.16
Nov.__14.69__10.26
Dec.__14.76__10.66

Find the test statistic t, the threshold from Table A-3 and determine if we should reject H0, which in matched pairs tests is always that mu1 = mu2.

Answers in the comments.

Test results and errors

We do the hypothesis testing because we cannot truly know what reality is, only the test result. If we reject the null hypothesis Ho, we did so because of strong evidence. If there is an error, it is a Type I error. If we set the error threshold at 90% confidence, we expect such errors about 10% of the time. If it is set at 95% confidence, then Type I errors should happen about 5% of the time and at 99% confidence, Type I errors should only happen about 1% of the time.

If we fail to reject H0, the only type of error we can make is called Type II error. The probability of such errors is trickier to compute and we will not work on this problem during this class.

binomcdp and continuity correction problems

Note: the functions binompdf and binomcdf from the TI-83 and TI-84 are available under slightly different names if you have the Excel spreadsheet program.

TI-83 or TI-84: binompdf(n, p, r) is the same as BINOMDIST(r, n, p, 0) in Excel.

TI-83 or TI-84: binomcdf(n, p, r) is the same as BINOMDIST(r, n, p, 1) in Excel.



Problems:

a) What is the probability of 20 or less successes in 30 independent trials when the probability of success on any one trial is .6?

b) What is the probability of 20 or less successes in 30 independent trials when the probability of success on any one trial is .65?

c) What is the probability of 20 or less successes in 30 independent trials when the probability of success on any one trial is .7?




d) What is the probability of 30 or more successes in 40 independent trials when the probability of success on any one trial is .8?

e) What is the probability of 30 or more successes in 40 independent trials when the probability of success on any one trial is .75?

f) What is the probability of 30 or more successes in 40 independent trials when the probability of success on any one trial is .7?

g) Optional for those with TI-83 calculators or Excel. Find np and nq for each problem and how close the approximations are.

Answers in the comments.

practice problems for binomial distribution

There are problems list at the end of the post through this link, with answers in the comments.

Notes on Bayesian probability

You can find notes on Bayesian probability in three posts from last term you can find through this link. Here are a few more practice problems, with answers in the comments.

A) A trait shows up in 20% of the population and the test has a 2% error rate. Find p(error given test positive) and p(error, given test negative).

B) A trait shows up in 10% of the population and the test has a 1% error rate. Find p(error given test positive) and p(error, given test negative).

Practice problems for homework due 7/6

Take the information of this incomplete contingency table with categories left and right in the columns and yes and no in the rows and fill in the rest of the table using the degrees of freedom.


____________left____right_____row totals
Yes___________25________________75
No____________________50_______
col. totals___90_____________________grand total

Use the information from the completed table to find the following probabilities, both as fractions and as percents rounded to the nearest tenth of a percent.

p-hat(Yes) =

p-hat(Left) =
p-hat(Left and Yes) =

p-hat(Left or Yes) =

p-hat(Left, given Yes) =

p-hat(Yes, given Left) =

State the following complementary sets without using the word NOT, using the categories from above.

NOT (Left) =

NOT (Left or Yes) =

NOT(Right and Yes) =

Answers in the comments.