2024 – Assume you need to build a confidence interval for a population mean within some given situation Naturally you must determine

Assume you need to build a confidence interval for a population mean within some given situation.  Naturally, you must determine whether you should use either the t-distribution or the z-distribution or possibly even neither based upon the information known/collected in the situation.  Thus, based upon the information provided for each situation below, determine which (t-, z- or neither) distribution is appropriate.

  Then if you can use either a t- or z- distribution, give the associated critical value (critical t- or z- score) from that distribution to reach the given confidence level.

 

99% confidence

a. 99% confidence

b. 95% confidence

90% confidence

c. 90% confidence

d. 99% confidence

2. A student researcher is interested in determining the average (µ) GPA of all FHSU students, in order to investigate grade inflation at regional universities.  The data below represent the GPA’s of thirty randomly selected FHSU students.

 

 2.75 2.55 3.95 1.74 2.66 3.10 2.41 1.57 2.12

 4.00 3.21 1.95 3.75 1.45 3.01 2.29 2.66 3.95

<

 2.32 3.44 2.07 0.62 2.72 3.55 3.92 3.41 2.14

 a.  How do you know that you will need to construct the confidence interval using a t-distribution approach as opposed to a z-distribution?

 

We want to construct the mean value confidence interval for the GPA’s with a 90% confidence level.

Point estimate

b. Determine the best point estimate (average) for the mean GPA.

c. Determine the critical t-value(s) associated with the 95% confidence level.

d. Determine the margin of error.

e. Determine the confidence interval.

Contextual meaning

In a sentence, interpret the contextual meaning of your result to part e above…that is relate the values to this situation regarding the mean GPA’s of all FHSU students.

 

Determine the two chi-squared (χ2) critical values for the following confidence levels and sample sizes.

a. 95% and n=30

<

b. 99% and n=20

We are also interested in estimating the population standard deviation (σ) for all FHSU student GPA’s.  We will assume that GPA’s are at least approximately normally distributed.  Below are the GPA’s.

 

2.75 2.55 3.95 1.74 2.66 3.10 2.41 1.57 2.12

<

4.00 3.21 1.95 3.75 1.45 3.01 2.29 2.66 3.95

2.32 3.44 2.07 0.62 2.72 3.55 3.92 3.41 2.14

Out to the right, construct a 95% confidence interval estimate of sigma (σ), the population standard deviation.

5. (Multiple Choice) A hypothesis test is used to test a claim.  On a right-tailed hypothesis test with a 1.39 critical value, the collected sample’s test statistic is calculated to be 1.45.  Which of the following is the correct decision statement for the test?

 

Null hypothesis

 A. Fail to reject the null hypothesis

 B. Reject the null hypothesis

 C. Claim the alternative hypothesis is true

 D. Claim the null hypothesis is false

        

        

6. (Multiple Choice) A hypothesis test is used to test a claim.  A P-value of 0.08 is calculated on the hypothesis test with a significance level set at 0.05.  Which of the following is the correct decision statement for the test?

 

 A. Claim the null hypothesis is true

 B. Claim the alternative hypothesis is false

Null hypothesis

 C. Reject the null hypothesis

 D. Fail to reject the null hypothesis

        

        

7. (Multiple Choice) Which of the following is not a requirement for using the t-distribution for a hypothesis test concerning μ.   

 A. Sample size must be larger than 30

 B. Sample is a simple random sample

 C. The population standard deviation is unknown

        

        

8. In an effort to promote healthy lifestyles, health screenings are given to employees of a large corporation.  In running a promotional trial, 84 out of the 150 people who work in one office for the corporation participate in the health screening.

 

        

 a. Is the above information sufficient for you to be completely certain that more than 50% of all employees of the corporation will participate in the health screening?  Why or why not?

 

b. In establishing a statistical hypothesis testing of this situation, give the required null and alternative hypotheses for such a test, if it is desired that more than 50% of the employees participate in the health screening.

 

c. Based on your answer in part b, should you use a right-tailed, a left-tailed, or a two-tailed test? Briefly explain how one determines which of the three possibilities is to be used.

 

d. Describe the possible Type I error for this situation–make sure to state the error in terms of the percent of employees in the corporation who will participate in the health screenings.

 

e. Describe the possible Type II error for this situation–make sure to state the error in terms of the percent of employees in the corporation who will participate in the health screenings.

 

f. Determine the appropriate critical value(s) for this situation given a 0.025 significance level.

g. Determine/calculate the value of the sample’s test statistic.

h. Detemine the P-value.

i. Based upon your work above, is there statistically sufficient evidence in this sample to support that more than 50% of employees will participate in the health screening?  Briefly explain your reasoning.

 

The mean score on a certain achievement test at the turn of the century was 74.  However, national standards have been implmented which may lead to a change in the mean score.  A random sample of 40 scores on this exam taken this year yeilded the following data set.  At a 10% significance level, test the claim that the mean of all current test scores is not the same as in 2000.

 

 

 85 77 74 88 89 66 0 70

 73 76 86 74 73 82 72 0

 82 82 80 76 87 76 77 67

 72 49 73 75 82 73 81 30

 58 75 72 89 76 18 72 74

       

a. Give the null and alternative hypotheses for this test in symbolic form.

Determine the value of the test statistic.

Determine the appropriate critical value(s).

Detemine the P value.

Is there sufficient evidence to support the claim that the mean achivement score is now different than 73?  Explain your reasoning.

 

10. Listed below are pretest and posttest scores from a study.  Using a 5% significance level, is there statistically sufficient evidence to support the claim that the posttest scores were the higher than the pretest scores?  Perform an appropriate hypothesis test showing necessary statistical evidence to support your final given conclusion.

 

 

11. Multiple Choice:

 For each of the following data sets, choose the most appropriate response from the choices below the table.

  Data Set #1   Data Set #2 

  x y   x y 

  0 19   10 100 

  1 15   14 33 

  2 13   18 124 

  3 12   24 160 

  4 7   27 65 

  5 0   32 117 

  6 -3   36 27 

  7 -4   40 150 

  8 -7   45 44 

 A. A strong positive linear relation exists   A. A strong positive linear relation exists  

 B. A strong negative linear relation exists   B. A strong negative linear relation exists  

 C. A curvilinear relation exists   C. A curvilinear relation exists  

 D. No linear relation exists   D. No linear relation exists  

12. Create a paired data set with 5 data points indicating strong (but not perfect) positive linear correlation.  Determine the correlation coefficient value for your data

 

x y

1 6

2 13

3 17

4 24

5 29

13. To answer the following, use the given data that contains information on the age of eight randomly female staff members at FHSU and their corresponding pulse rate.

 

  Age (years) Pulse Rate (bpm)     

  42 98     

  34 80     

  49 98     

  27 63     

  42 84     

  18 49     

  41 80     

  21 55     

        

        

b. Based on the scatterplot, does it look like a linear regression model is appropriate for this data?  Why or why not?

c. Add the line-of-best fit (trend line/linear regression line) to your scatterplot. Give the equation of the trend line below.  Then  give the slope value of the line and explain its meaning to this context.

 

d. Determine the value of the correlation coefficient.  Explain what the value tells you about the data pairs? 

e. Does the value of the correlation coefficient tell you there is or is not statistically significant evidence that correlation exists between the age and pulse rates of female staff members?  Explain your position.  (HINT: application of table A-6 is needed!)

 

 

f. Based on the above, what is the best predicted pulse rate of a 30 year old female staff member?

14. Complete the following in regard to the hair color category of the data set listed to the right.

b. Directly below, create a statement about “hair color” that the information from part a supports or suggests. Remember this statement is only to be made based on the sampled students’ data for the online course as shown in your work in part a.

 

c. Next, create a new statement/claim about “hair color” in regard to all students at FHSU  (this claim does not have to match the information in the bar chart).  Then briefly discuss whether the chart proves or disproves your statement.

 

15. Complete the following in regard to the footlength and height categories of the data set listed to the right.

         

 a. In the region to the right, produce a scatterplot of the height versus footlength data (remember this means footlength runs along the horizontal axis as the independent variable and height along the vertical axis as the dependent variable.)  Based upon your scatterplot, briefly discuss below your thoughts on whether the “visual” trend between the individuals’ footlength and height appears linear, curvilinear, or has no general trend at all.

 

 

 

 

b. Complete the following:

        

 i. Include the trend line’s graph and equation on the scatterplot created in part a.  Give the line’s equation below and explain within this context what the “x” and “y” variables represent in the equation.

 

ii. Below, explicitly state the slope of your trend line and discuss what the value of the slope signifies in terms of this context. 

 

c. Determine the value of the correlation coefficient for this paired data.  Explain what this value tells you and determine if it implies that there is/is not statistically significant correlation between the two variables of height and footlength. (Recall the need to use table A:6 from the text in answering the last portion of this question.

 

 

d. Using the predicition equation from part bi. above, predict the height of an individual whose footlength is 23.45 cm.

        

 

e. Finally, critique the statement “since the correlation coefficient is statistically significant then this means that having a long foot causes one to be tall.”  Specifically, address the issue of “causation” in relation to significant statistical correlation