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Elements of Statistics
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.
a. 99% confidence
d. 95% confidence
c. 90% confidence
e. 99% confidence
2. The data below shows the birth weights (in kilograms) of thirty randomly chosen male babies born in Hays Medical in past year. It is also known that the population standard deviation of birth weights for all male babies born is 0.0731 kg (based on data from the New York State Department of Health).
3.73 4.37 3.73 4.33 3.3 3.39 3.68 4.68 3.72 3.02
3.96 2.21 2.67 4.09 3.02 2.76 3.67 3.76 3.45 2.54
2.55 3.44 3.07 4.23 2.92 3.55 3.92 3.41 4.14 3.15
a. How do you know that you will need to construct the confidence interval using a z-distribution approach as opposed to a t-distribution?
We want to construct the mean value confidence interval for all Hays male babies' birth weights with a 99% confidence level.
b. Determine the best point estimate (average) for the mean birth weight.
c. Determine the critical z-value(s) associated with the 99% confidence level.
d. Determine the margin of error.
e. Determine the confidence interval.
f. 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 birth weights of all Hays male babies born.
There is a 99% chance that the true population mean birth weight is within the limits of the confidence interval calculated in part (e). This is because 99 out of 100 similarly constructed confidence intervals (i.e. same confidence level, same sample size) would be expected to contain the true population mean birth weight.
3. Determine the two chi-squared (χ2) critical values for the following confidence levels and sample sizes.
a. 98% and n=25
b. 90% and n=60
4. We are also interested in estimating the population standard deviation for all male babies birth weights (in kilograms). We will assume that birth weights are at least approximately normally distributed. Below are the birth weights of 30 randomly chosen male babies from Hays Medical.
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 1.41. Which of the following is the correct decision statement for the test?
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 0.001 is calculated on the hypothesis test with a significance level set at 0.01. 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
C. Reject the null hypothesis
D. Fail to reject the null hypothesis
9. The mean score on a certain achievement test at the turn of the century was 73. However, national standards have been implmented which may lead to a change in the mean score. A random sample of 48 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 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
60 88 20 99 50 35 78 66
a. Give the null and alternative hypotheses for this test in symbolic form.
b. Determine the value of the test statistic.
c. Determine the appropriate critical value(s).
d Detemine the P-value.
e. Is there sufficient evidence to warrant rejection of the claim that the mean achivement score is now is 73, the same as in 2000? 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.
PreTest PostTest Difference (Post - Pre)
24 25 1 Average Difference 2.428571429
11 18 7 Std. Dev. Of Differences 2.572750983
14 16 2
25 29 4
17 16 -1
28 29 1
22 25 3
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
0.3 3790 -1 -4
0.4 3354 -2 -10
0.5 2986 2 5
0.6 2613 3 4
0.7 2277 -3 -19
0.8 1765 6 -10
0.9 1343 7 -20
1 1151 -1 -4
1.1 510 0 1
12. Give a real life example of two variables that are likely to be negatively correlated. Specifically explain why you believe they are negatively correlated.
13. To answer the following, use the given data set for lengths (in inches) and corresponding weights (in pounds) of randomly selected black bears captured in the backcountry of Colorado
lengths (inches) weights (pounds)
40 65
64 256
65 216
49 94
47 86
59 189
61 202
49 102
a. Construct a scatterplot for this data set in the region to the right (length as the independent variable, and weight as the dependent.)
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 length and weight of black bears? Explain your position. (HINT: application of table A-6 is needed!)
f. Based on the above, what is the best predicted weight of a bear with a length of 45 inches?
