Inference on Proportions (one proportion and two ...



Inference on Proportions (one proportion and two proportions) Chapters 19 and 20

τ Notation: p = pop’n proportion n = sample size [pic] = sample proportion

τ Sampling dist’n of [pic] and Sampling dist’n of [pic]

Shape

Mean

Standard deviation (also called Standard Error or SE)

Conditions:

• SRS

• Large population (at least ten times as large as the sample)

• Appropriate sample size to assure that the shape is approximately normal:

One proportion: [pic] and [pic]

Two proportions: [pic] and [pic]

and [pic] and [pic]

τ Simulation to explore sampling distributions.

Notation: n is the individual sample size. N is the number of replications – use at least 1000.

Use this applet:

τ What is the SE for each type of parameter?

|[pic] where [pic]known |[pic] where |[pic] |[pic] |[pic] |

| |[pic]NOT known | | | |

|[pic] |[pic] |[pic] | | |

| | |Depends on unknown p, so we must |Generalize from one |Generalize from one |

|Pop’n standard deviation |Pop’n standard deviation |estimate it. Slightly different |mean |proportion |

|given. |estimated by sample standard|estimate in confidence interval and | | |

| |deviation. |hypothesis test | | |

| | |and for computation of sample size. | | |

τ Confidence Intervals

|General | [pic] where [pic]known |[pic] |[pic] |

|[pic] |[pic] | | |

| | | | |

τ Hypothesis Testing

|General | [pic] where |[pic] |[pic] |

| |[pic]known | | |

| | | | |

|[pic] |[pic] | | |

| | | | |

|Find P-value from z-score. Use it to write the | | | |

|conclusion in context. | | | |

τ Sample size needed to obtain a given margin of error

|General | [pic] where [pic]known |[pic] |

|Take margin of error formula and solve|[pic] | |

|for n. | | |

| | | |

| | | |

τ Conditions

Tradeoff between:

1. Conceptual Understanding:

a. Easy to understand

b. Easy to remember / look up

c. Easy to remember how to check

2. Make the most of the data you have.

a. When is it OK to use the method on data that don’t exactly meet the conditions?

b. How close do the data have to come to meeting the conditions?

c. Is there a different method that will work for data that don’t meet the main conditions? If so, be aware that it exists and can be used.

τ Our class: What we’ll use: Inference on means

Learn to make the most of the data you have. Use the conditions in the “Robustness” sections. (In particular, the three for [pic], [pic], and [pic] and, for the two-sample t-test, the “sum of two sample sizes” guidelines.)

τ Our class: What we’ll use: Inference on proportions

We’ll focus on conceptual understanding. So, instead of using the sample-size conditions listed in the text, we’ll just use the ones with 10 listed in the description of the sampling distributions of [pic] and [pic] in this handout.

Also, know that if your data do not meet these sample size conditions, there is a very similar method that one can use to find Confidence Intervals. That’s the “plus-four” method, which is also called “Accurate confidence intervals.” Read about it, but I will not expect you to find confidence intervals using it.

τ Using software for computations

Learn to use software to do these computations. Then you can do enough problems to really learn the concepts without going crazy doing computations!!

So, what will I test you over? See below.

Tests 4 and 5:

Lots:

Identify which type of problem each is and how to set it up, including discussing conditions, describing the parameters in words, etc. Then take a given computed value of the statistic and use that to complete the problem, including writing the conclusion in context.

Some:

Write the appropriate formula and plug in the appropriate numbers from the given data.

A few:

Carry out the computations by hand.

Test 5. MAYBE . I might assign something to do on software on Test 5, since we’ll take that in class.

Real life:

Your tasks:

• Formulate the question.

• Identify, collect, or produce appropriate data to help answer the question.

• Select an appropriate method to analyze the data, including checking conditions.

• Usually use software to make the appropriate graphs and do the appropriate computations.

• Interpret the results in context.

• Write reports or be able to explain your results to your supervisor or clients.

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