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1. For One Sample Test Around the Mean.

1.1    Calculating Confidence Intervals 

A Confidence Interval is an interval or range of numbers.  It is calculated around a "point estimate" (such as the mean of a group of numbers). We calculate confidence intervals for continuous numbers because we can not estimate one number out of infinitely.  We use confidence intervals to estimate how far the point estimate is from the parameter of a population of a specific level of confidence.

Large Sample Confidence Intervals for a Population Mean

Z Statistic A statistic having the standard normal (or Gaussian) distribution. It arises in test of a mean which is compared to a known variance.

  1. Large Sample > or = to 30

  2. Use when population variances are known or unknown

  3. Sampling From Independent Normal Distributions 

  4. One Sided or Two Sided test 

  5. One mean/average known

The following formulas are for two sided test.  Confidence intervals come with two sides.

You may also use the t test formula if population variance is unknown.

Small Sample Confidence Intervals for a Population Mean

Degrees of Freedom Are the number of independent elements that go into a statistic. An independent comparison. Number of pieces of information that are independent of one another in that they cannot be deduced from one another.  The t test requires one to use Degrees of Freedom with looking up the correct values in the t tables.

When the level of confidence and sample standard deviation remain the same, a confidence interval for a population mean based on a sample of n = 100 will be narrower than a confidence interval for a population mean based on a sample of n = 50.

Information created by Craig Stevens Based on material found in a couple of different texts.

Bottom more appropriate for confidence intervals (two sided in nature).

Determine the Sample Size 

B = the value farthest away from the population that you will allow the sample mean to be.  the "n" will be the size of the sample required for us to be confident (to some degree, i.e. 95%) of our answer.  

 

Confidence Intervals for Parameters of Finite Populations

  • point estimate = a one-number estimate of the value of a population parameter 
  • population parameter = a descriptive measure of the population) 
  • p = population proportion 
  • p hat = sample proportion
  • When the sample size and the sample proportion remain the same, a 90% confidence interval for a population proportion (p hat) will be narrower than the 99% confidence interval for p.  The bell shaped curve will have narrower distance between the standard deviations.
  • When the level of confidence and sample proportion remain the same, a confidence interval for a population proportion (p hat) based on a sample of n = 100 will be wider than a confidence interval for p based on a sample of n = 400
  • When the level of confidence and sample size remain the same, a confidence interval for a population proportion p will be wider when is larger than when it is smaller.

 

Determining Sample Size for Confidence Interval for p

 

B = the value farthest away from the population that you will allow the sample mean to be.  the "n" will be the size of the sample required for us to be confident (to some degree, i.e. 95%) of our answer.  

        p = population proportion 

Confidence Intervals for Population Mean and Total for a Finite Population

For a large (n > or = to 30) random sample of measurements selected without replacement from a population of size N.

Confidence Intervals for Proportion of and Total Number of Units when Sampling a Finite Population

For a large random sample of measurements selected without replacement from a population of size N

 

1.2    Hypothesis Testing

Null and Alternative Hypotheses

Ho = The null hypothesis, is a statement of the basic proposition being tested. The statement generally represents the status quo and is not rejected unless there is convincing sample evidence that it is false.

Ha = The alternative or research hypothesis, is an alternative (to the null hypothesis) statement that will be accepted only if there is convincing sample evidence that it is true.

Six Steps of Hypothesis Testing 

  1. Determine null and alternative hypotheses 
  2. Specify level of significance (probability of Type I error) ? 
  3. Select the test statistic that will be used. 
  4. Collect the sample data and compute the value of the test statistic. 
  5. Use the value of the test statistic to make a decision using a rejection point or a p-value. 
  6. Interpret statistical result in (real-world) managerial terms

 

Examples From our Textbook

Large Sample Tests about a Mean: Testing a One-Sided Alternative Hypothesis

If the sampled population is normal or if n is large, we can reject H0: m = m0 at the a level of significance (probability of Type I error equal to a) if and only if the appropriate rejection point condition holds.

Large Sample Tests about Mean: p-Values

The p-value = the observed level of significance =  the probability of observing a value of the test statistic greater than or equal to z when Ho is true.  It measures the weight of the evidence against the null hypothesis and is also the smallest value of alpha for which we can reject Ho.

Large Sample Tests about a Mean: Testing a Two-Sided Alternative Hypothesis

Small Sample Tests about a Population Mean 

Hypothesis Tests about a Population Proportion 

Type II Error Probabilities and Sample Size Determination 

The Chi-Square Distribution 

Statistical Inference for a Population Variance

 

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