Let x = age in years of a rural Quebec woman at the time of her first marriage. In the year 1941, the population variance of x was approximately σ² = 5.1. Suppose a recent study of age at first marriage for a random sample of 41 women in rural Quebec gave a sample variance s² = 2.6. Use a 5% level of significance to test the claim that the current variance is less than 5.1. Find a 90% confidence interval for the population variance.

(a) What is the level of significance?


State the null and alternate hypotheses.

H_o: σ² = 5.1; H₁: σ² > 5.1H_o: σ² < 5.1; H₁: σ² = 5.1    H_o: σ² = 5.1; H₁: σ² ≠ 5.1H_o: σ² = 5.1; H₁: σ² < 5.1

(b) Find the value of the chi-square statistic for the sample. (Round your answer to two decimal places.)


What are the degrees of freedom?


What assumptions are you making about the original distribution?

We assume a uniform population distribution.We assume a binomial population distribution.    We assume a exponential population distribution.We assume a normal population distribution.

(c) Find or estimate the P-value of the sample test statistic.

P-value > 0.1000.050 < P-value < 0.100    0.025 < P-value < 0.0500.010 < P-value < 0.0250.005 < P-value < 0.010P-value < 0.005

(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis?

Since the P-value > α, we fail to reject the null hypothesis.Since the P-value > α, we reject the null hypothesis.    Since the P-value ≤ α, we reject the null hypothesis.Since the P-value ≤ α, we fail to reject the null hypothesis.

(e) Interpret your conclusion in the context of the application.

At the 5% level of significance, there is insufficient evidence to conclude that the variance of age at first marriage is less than 5.1.At the 5% level of significance, there is sufficient evidence to conclude that the that the variance of age at first marriage is less than 5.1.    

(f) Find the requested confidence interval for the population variance. (Round your answers to two decimal places.)

Interpret the results in the context of the application.

We are 90% confident that σ² lies outside this interval.We are 90% confident that σ² lies above this interval.    We are 90% confident that σ² lies below this interval.We are 90% confident that σ² lies within this interval.

In clinical trials of the allergy medicine Clarinex (5 mg), it was reported that 50 out of 1655 individuals in the Clarinex group and 31 out of 1652 individuals in the placebo group experienced dry mouth as a side effect of their respective treatments. Test the hypothesis that a greater proportion of individuals in the experimental group experienced dry mouth compared to the individuals in the control group at the α=0.01 level of significance.

Null Hypothesis:

Alternate Hypothesis:

P-value:

Conclusion:

Interpretation:

Construct a 95% confidence interval for the difference between the two population proportions Experimental – Control. Explain what it means.

Problem 28. A fair six-sided die is rolled repeatedly and the rolls are recorded. When two consecutive rolls are identical, the process is ended. Let S denote the sum of all the rolls made. Is S more likely to be even, odd or just as likely even as odd?

I am working on creating a Wiebull distribution from a large set of data I have. Everything I find online says that I should be given the shape parameter (beta), and scale parameter (eta/apha). I do not have these numbers and I am not sure how to find them to accurately create a Weibull dist.

1. When discussing the concept of risk, what type of outcomes are we considering (negative outcomes or positive outcomes)?   (For example, would we talk about the risk of DYING from heart failure OR the risk of SURVIVING heart failure?)

2. If the risk of Outcome A is 154% compared to the risk of Outcome B, has the risk for Outcome A increased, decreased or stayed the same?

3. What decimal (e.g x.xx) would you use to express a risk of 67%?  Has the risk increased, decreased or stayed the same?

4. When considering the confidence interval surrounding a risk (e.g. risk of 0.84 with a 95% CI of 0.65 – 0.98), what is the “key” number to look for, and why?  What is the relationship between statistical significance and this “key” number?

5. In a population of cigarette smokers, 35 out of 1,436 will develop small cell lung cancer (SCLC).   What is the risk of developing SCLC in this population?

6. In a population of non-smokers, 188 out of 153,678 will develop SCLC.   What is the risk of developing SCLC in this population?

7. What is the relative risk for developing SCLC among smokers as compared to non-smokers?

8. In your own words, what is the difference between risk and odds?

9. Construct a 2x2 table for the statistics given for SCLC above and calculate the odds ratio.

10. In your own words, what is the meaning of the odds ratio? (How would you explain this to a patient?)        

11. If the risk for developing a neural tube defect (NTD) without folate supplementation is 0.63% and the risk for developing a NTD with folate supplementation is 0.03%, what is the relative risk for not taking folate, and what is the relative risk reduction for taking folate?

12. What is the absolute risk reduction for taking folate supplementation?  What is the NNT for taking folate supplementation?

REVIEW QUESTIONS

1.   When does a manufacturer apply for an IND?

2.   When does a manufacturer apply for an NDA?

3.   After what stage of the drug development process is the drug released to market?

4.   A study contains the statement, “Based on the p-value of 0.056, the null hypothesis was accepted and it was determined that the diabetic control on drug A was not different from diabetic control on drug B.”   What branch of statistics is this statement representing?

5.   By definition the 50^thpercentile is also known as what”

6.   On a stem and leaf plot, the range is determined by taking the largest possible valuable and subtracting the lowest possible value. True or False?

7.   What part of a box plot represents the interquartile range?

8.   Between a pie chart, bar chart and histogram, which two are used for discreet data?

This question has several parts that must be completed sequentially. If you skip a part of the question, you will not receive any points for the skipped part, and you will not be able to come back to the skipped part.

Test the null hypothesis that the slope is zero versus the two-sided alternative in each of the following settings using the  α = 0.05  significance level.

(a)n = 20, ŷ = 24.5 + 1.6x, and SE_b1 = 0.75

(b)n = 30, ŷ = 30.1 + 2.7x, and SE_b1 = 1.35

(c)n = 100, ŷ = 29.4 + 2.7x, and SE_b1 = 1.35

Question 1

A residual is:

choose one

• The difference between a data point and the regression line.

• A value that can be 1 or zero.

• A value that is always negative because it is a difference

• The difference between two different lines.

Question 2

The correlation coefficient:

choose one

• Is a number with a range from -1 to 1

• If there is no correlation, the coefficient is negative

• If the correlation coefficient is negative, it indicates a strong positive relationship between x and y

• All of the above

Question 3

The assumptions we use to determine the validity of predictions include:

choose one

• For every specific value of y, the value of x must be normally distributed about the regression line.

• The sample was collected carefully

• The standard deviation of each dependent variable must be the same for each independent variable

• All of the above

Question 4

A positive straight line relationship:

choose one

• Show no change in the variables

• Show that both variables increase in value

• Shows that as the values of x increases, the values of y decreases

• Slopes down

Question 5

Because some people are unable to stand to have their height measured, doctors use the height from the floor to the knee to approximate their patients’ height (in cm).

a. Use Excel to determine the correlation coefficient of this data

b. Use Excel to determine the regression equation of this data

c. Find the overall height from a knee height of 45.3 cm

d. Find the overall height from a knee height of 52.7 cm

Choose one

• a. r = 0.73220213

  b. Equation: y = 2.0217x + 67.746

  c. 159.32901

  d. 174.28959

• a. r = 0.82544241

  b. Equation: y = 2.5109x + 40.79

  c. 154.53377

  d. 173.11443

• a. r = 0.53611996

  b. Equation: y = 2.0217x + 67.746

  c. 159.32901

  d. 174.28959

• a. r = 0.908553861

  b. Equation: y = 2.5109x + 40.79

  c. 154.53377

  d. 173.11443

Question 6

The coefficient of determination:

Choose one

• Represents the percentage of the data that can be explained by the correlation

• Is equal to the ratio of the explained variation to the total variation

• Is calculated by squaring the correlation coefficient.

• All of the above

Question 7

A simple regression model uses a straight line to make predictions about future events.

Choose one

• True

• False

Question 8

Once we have a simple regression line, we can use it to predict values for the independent variable X and the dependent variable Y.

Choose one

• True

• False

Question 9

The independent variable is represented by a y.

Choose one

• True

• False

Question 10

Outliers:

Choose one

• Greatly affect the value of r

• Should be identified and taken out of the data before any correlation analysis

• Are easily identified in a scatterplot

• All of the above

Which of the following is not a characteristic for a normal distribution?

a. It is a symmetrical distribution

b. The mean is always zero

c. The mean, median, and mode are all equal

d. It is a bell-shaped distribution

e. The area under the curve always equals 1.0

The following table summarizes the results of a two-factor ANOVA evaluating an independent-measures experiment with 2 levels of factor A, 3 levels of factor B, and n = 6 participants in each treatment condition.

A. Fill in all missing values in the table. Show your work (i.e., all computational steps for finding the missing values). Hint: start with the df values.

B.  Do these data indicate any significant effects (assume p < .05 for hypothesis testing of all three effects)?

           Note: in order to receive full credit, your answer for each effect should include:

• the null hypothesis H₀ & the alternative hypothesis H₁
• the critical F value used for the decision about H₀
• if the difference is statistically significant, the computed effect size, η²
• the conclusion in APA style format

                      Source                 SS          df            MS                

          Between Treatments 75 ____

               Factor A                    ____       ____      ____     F_A = ___   

               Factor B                    ____       ____ 15 F_B = ___

               A x B    ____     ____      ____      F_AxB = 6.00

          Within Treatments           ____       ____             

          Total    165          ____

An obstetrician wants to know whether or not the proportions of children born on each day of the week are the same. She randomly selects 500 birth records and obtains the data shown in table. Is there reason to believe that the day on which a child is born occurs with equal frequency at the alpha= 0.01.

PLEASE FOLLOW THE STEPS BELOW:

• You need formulate null hypothesis and alternative hypothesis
• Select alpha
• Compute expected count
• Compute chi-square
• Find p-value
• Make a decision.

A recent study found that toddlers who have a diet high in processed foods may have a slightly lower IQ later in life. The conclusion came from a long-term investigation of 14,000 people whose health was monitored at 3,4,7, and 8 years of age.

e) One analysis found that of the 4000 children for which there were complete data, there was a significant difference in IQ between those who had had “processed” (i.e., junk) food and those who followed health-conscious diets in early childhood. Is this an experiment? Why or why not?

f) Discuss at least two explanatory factors that could conceivably confound the relationship between diet and IQ.

Ross Simons, a jewelry cataloguer, has tested two creatives for the cover of the spring 2018 catalogue. The themes were inspired for the birthstones of April and May. The timing of the drops for the two executions was March 15, 2017. The results are shown below and based on the number of orders from each test cell. The interest is in whether the cover, which is the only difference in the two catalogues, can be associated with a difference in average order size.

Construct a 95% confidence interval for the difference between the average order size in dollars for each cell. Remember to then fully interpret results including a recommendation for the business decision on the indicated strategy.

One measure of the value of a stock is its price to earnings ratio (or P/E ratio). It is the ratio of the price of a stock per share to the earnings per share and can be thought of as the price an investor is willing to pay for $1 of earnings in a company. A stock analyst wants to know whether the P/E ratios for three industry categories differ significantly. The following data represent simple random samples of companies from three categories: (1) financial, (2) food, and (3) leisure goods.

1. Test the null hypothesis that the mean P/E ratio for each category is the same at the alpha = 0.05 level of significance with the following steps:

• Formulate the null hypothesis and alternative hypothesis.

H₀: ______________________________ versus   H₁: _________________________________________.

• Check the assumptions and find MST, MSE, and the test statistic of fo with degrees of freedom of numerator and denominator.
• Make a decision and draw a conclusion with alpha = 0.05 level of significance.

2. If the null hypothesis is rejected in part (a), use Tukey’s test to determine which pairwise means differ using a familywise error rate of alpha = 0.05, where qa,v,k = q0.05,12,3 = 3.773 from the table of critical values for Tukey’s Test.

H₀: __________________ versus   H₁: ____________________.

q0=

Decision:

H₀: __________________ versus   H₁: ____________________.

q0=    

Decision:

H₀: __________________ versus   H₁: ____________________.

q0=

Decision:

• Use lines to indicate which population means are not significantly different.

Needing to measure how each factor (WAR, ERA, WHIP) effects salaries. Needing to see the probability an individual with a higher WAR will also receive a larger paycheck versus a person has a lower WHIP or ERA (these the lower the better). For example the out of the top 10 lowest WHIP 6 are in the top 33% of payed players because WHIP being low is a good factor. I believe the best way to show this would be using either t or z distribution formulas. Any help would be greatly appreciated. Looking to see how the different variables of WAR (wins above replacement) (this being higher should be a benefit when talking about pay increases), WHIP (walks plus hits per inning pitched) (the lower the WHIP the better when talking about pay increases), and ERA (earned run average) (the lower the better when talking about pay increases) effect the salaries of the players. Needing to know which effects the pay the most and which the least and by how much percentage wise and how this is done. Trying to show that having a higher WAR correlates to a higher salary and having a lower ERA and WHIP correlates to a higher salary and which has a greater impact in salary. Below are the top 30 base salaries with their average statistics to support their pay.