# Lab Activity 6 – Confidence Intervals – Proportion and One Mean

August 30, 2017

Question
Lab Activity 6 – Confidence Intervals – Proportion and One Mean

1. The term sampling frame refers to the group that actually had a chance to get into the sample. Ideally, this is the same as the population of interest, but sometimes it isn’t. In the following situation, describe the population, the sampling frame, the sample, the parameter of interest, and the statistic.

In 2010, the Centers for Disease Control and Prevention (CDC) collected a survey of the hospital records of approximately 14,000 individuals who were discharged from 239 hospitals across the United States. As part of the survey, data were collected on the proportion of newborn boys who received a circumcision. The purpose was to find the proportion of newborn American males that receive circumcisions. The survey found that 58% of newborn American males received circumcisions.

a. Population =

b. Sampling frame =

c. Parameter =

d. Sample =

e. Statistic =

2. From the Data Sets folder open the OlympicSwimming2012 data. This is a dataset of all swimmers in the 2012 Olympic games that won a medal.

Do you think this dataset is a representative sample for all swimmers in the 2012 Olympics? Why or why not?

What type of sampling is being used if we are interested in the population of all swimmers?

Referring to condition regarding using the normal approximation for sample proportions (i.e. is np-hat ? 15 and n(1-p-hat) ? 15) verify that this condition has been met thus allowing us to use the normal approximation techniques. To find p-hat, you would add the total number of 1s in column C4 (“Is American?”) and divide it by n: p-hat=(# of Americans)/n. Note: We could also check the above assumption using an identical statement: The number of individuals that fall into the category of interest – American, in this case – is at least 15. The number that do not fall into the category of interest is also at least 15.

Use Minitab or SPSS to calculate a 95% one-sample proportion confidence interval for the percentage of swimmers who are American. What is your interval?

Minitab Users: Go to Stat > Basic Statistics > 1 Proportion. Put the variable Is American? In the box. Click Options, and in the “Method” drop-down menu, select “Normal Approximation”. Click “OK”, and your output should be displayed.

Minitab Express Users: Click Statistics > One Sample > Proportion. Put the variable Is American? In the “Sample” box. Click Options, and in the “Method” drop-down menu, select “Normal Approximation”. Click “OK”, and your output should be displayed.

SPSS Users: Go to Analyze > Compare Means > One Sample T Test. Select the variable “Is American?” and move it to the Test Variables field. Click ‘Options’ and enter 95 for the confidence level percentage. Click Continue then OK.

f. For the interval, answer the following:

What is the sample proportion, p-hat (also known as the point estimate)?

What is the z multiplier used for your interval?

What is the standard error?

What is the margin of error?

f. If the confidence level in part g were changed to 85% would your resulting interval be wider or narrower?

g. Say someone was to claim that 50% of the swimmers who won medals in the 2012 Olympic games were American. Based on your confidence interval do believe that this percentage is reasonable, too high, or too low and explain why.

3. For mean confidence intervals we call into use the T-Table which can be found in this week’s folder. The first concept to understand is the idea of Degrees of Freedom (DF). For our activity today, since we are only concerned with one mean (either from one sample of a difference between paired data), DF = sample size – 1 (i.e. n – 1). When finding confidence intervals, the T-Table provides the t multiplier (t*) for the confidence interval expression:

x ? ±t^*× s/?n if data is consists of only one sample

NOTE: you will notice that the DF column in this tables only increases by 1 from 1 to 30. After that the increments vary. If your DF is NOT found in the table then conservatively use the CLOSEST DF in the table that does not exceed the DF of interest. For example, if the degrees of freedom were 37 then from the table use 30.

Find t-multipliers and DF from T-Table for the following conditions:

Confidence Level 90%, n = 20: t* = DF =

Confidence Level 95%, n = 20: t* = DF =

Confidence Level 95%, n = 43: t* = DF =

Confidence Level 99%, n = 43: t* = DF =

What do you notice that happens to t* as the level of confidence decreases for the same sample size?

What do you notice that happens to t* as sample size decreases for the same level of confidence?

4. Using the OlympicSwimming2012 data, let’s estimate the true age for all Olympic swimmers (not just the ones that won medals). Assuming that our sample represents this population, calculate a 1-mean 95% confidence interval to estimate the parameter. First we will do by hand and then using software. To start, the sample mean, or point estimate, (x ?) is 23.339 and n = 124 and s = 3.806.

a. Calculate the standard error of the mean.

b. What are the DF and t* from the T-table?

DF = t* =

c. Calculate the 95% Confidence Interval and provide an interpretation of this interval.

x ? ±t^*× s/?n

d. Now use software to verify your results by calculating a 95% one-sample T confidence interval. Copy and paste your output results below. Do your results by hand and those from software roughly match?

Minitab Users: Go to Stat > Basic Statistics > One Sample t. Select the Age variable and move it to the Samples in Columns field. Click Options to verify that the confidence level is correct (95). Click OK.

Minitab Express Users: Go to Statistics > One Sample > t. Double-click the Age variable to move it into the “Sample” box. Click “Options”, and verify that the confidence level is 95. Click OK.

SPSS Users: Go Analyze > Compare Means > One Sample T Test. Select the variable “Age” and move it to the Test Variables field. Click ‘Options’ and make sure 95 is entered for the confidence level percentage. Click Continue then OK.

e. Write a sentence that explains what this interval means.

f. Based on the interval calculated, if someone claimed that the true mean age for the 2012 Olympic swimmers was 25 would you believe them? Explain.

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