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.

A Gallup Poll is done using random digit dialing to reach individuals in households with land-line telephones. The purpose is to estimate the proportion of U.S. adults who favor stronger gun control laws. One-thousand persons are sampled, and 63% favor stronger gun control.

1. Population =
2. Sampling frame =
3. Parameter =
4. Sample =
5. Statistic =

2 From the Data Sets folder open the Class Survey data and calculate a 90% one-sample proportion confidence interval for the percentage of students who smoke cigarettes.

1. Since this is a class survey of stat200 students at PSU what do you believe is the best population that this data represents?
2. How is the class survey representative of that population?
3. How would you best describe the sampling technique used in attaining the Class Survey?
4. Referring to condition regarding using the normal approximation for sample proportions (i.e. is n \[\hat{p}\] ≥ 15 and n(1- \[\hat{p}\] ) ≥ 15) verify that this condition has been met thus allowing us to use the normal approximation techniques (e.g. Use normal approximation option in Minitab or the z-multiplier if doing by hand)
5. Use Minitab or hand if using SPSS to calculate a 90% one-sample proportion confidence interval for the percentage of students who smoke cigarettes Remember to check Option " Use test and interval based on Normal Distribution " if using Minitab or the z-multiplier if calculating by hand . What is your interval?
6. For the interval, answer the following:
   What is the sample proportion, \[\hat{p}\] ?
   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 95% would your resulting interval be wider or narrower?
7. The U.S. Government reported that 23% of US adults age 18-24 smoked cigarettes. Based on your confidence interval do believe that this percentage is reasonable, too high, or too low for Penn State students and explain why.

3 For mean confidence intervals we call into use 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, 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 = 25: t* = 1.711 DF = 24

Confidence Level 95%, n = 25: t* = 2.064 DF = 24

Confidence Level 95%, n = 35: t* = 2.032 DF = 34

Confidence Level 99%, n = 35: t* = 2.728 DF = 34

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

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

4 Using the Class Survey data, PSU wants to estimate the true SATM scores for its undergraduate population. Assuming that our survey 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 ( ) is 599 and n = 216 and s = 85.34.

1. Calculate the standard error of the mean.
2. What are the DF and t* from T-table ?
   DF = t* =
3. Calculate the 95% Confidence Interval and provide an interpretation of this interval.
   
4. 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?
5. Write a sentence that explains what this interval means.
6. Based on the interval calculated, if Penn State claimed that the true mean SATM score for the 2005 incoming freshman was 610 would you believe them? Explain.

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Solution: The downloadable solution consists of 6 pages, 1023 words and 5 charts.
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