# Lab Activity 6 Confidence Intervals

Question

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.

If I, your instructor, wanted to conduct a survey to find the average grade obtained in Stat 200 for PSU students in the fall 2014 semester, I could look at the average grade received for a random sample of my students’ grades at the end of a given semester. For the Fall 2014 semester, a random sample of 34 of my students had an average overall grade of 82.665%. Tell me what the following terms are in the context of this (tiny) analysis.

a. Population =

b. Sampling frame =

c. Parameter =

d. Sample =

e. Statistic =

2.

From the Data Sets folder open the Final_Grade data. This is the data set described in the question above.

a. Do you think this data set is a representative sample for all students who took Stat 200 last semester?

b. Why or why not is this sample representative of all Stat 200 students?

c. What type of sampling is being used if we are interested in the population of all Stat 200 students?

d. Referring to condition regarding using the normal approximation for sample proportions (i.e. is n*p-hat ? 10 and n*(1-p-hat) ? 10) verify that this condition has been met thus allowing us to use the normal approximation techniques (e.g. Use normal approximation option in Minitab Express or the z-multiplier if doing by hand)

e. Use Minitab Express to calculate a 95% one-sample proportion confidence interval for the percentage of Stat 200 students who are male. What is your interval?

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

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?

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

h. Say someone was to claim that 70% of my students last semester were male. 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:

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 = 16: t* = DF =

Confidence Level 95%, n = 16: t* = DF =

Confidence Level 95%, n = 38: t* = DF =

Confidence Level 99%, n = 38: 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 Final_Grade data, let’s estimate the true overall grade for all Stat 200 students. 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, ( ) is 82.665 and n = 34 and s = 13.107.

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.

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? Remember to paste your software output.

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

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

f. Based on the interval calculated, if someone claimed that the true overall grade for all Stat 200 students was 90 would you believe them? Explain.

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