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The mean height of a certain kind of plant is 191 centimeters. Suppose we want to carry out a hypothesis test to see if the mean height when these plants are treated with a certain chemical differs from 191. State the null hypothesis and the alternative hypothesis that we would use for this test # D Oca OSD

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Introductory Statistics
OER Edition, 2013
Openstax
ISBN: 9781938168208

Related questions

(14 marks) Fortune magazine publishes a list of the world’s billionaires each year. We will examine the top 97 of the 1992 list. Their wealth, age, and geographic location (Asia, Europe, Middle East, United States, and Other) are reported. Wealth is a right-skewed variable, so we will consider the natural logarithm log wealth to be the response and will see if age and geographic location are predictors.
(a) (2 marks) Let
Ebe a variable that is 1 if the person is from Europe and 0 otherwise,
Mbe a variable that is 1 if the person is from the Middle East and 0 otherwise,
Obe a variable that is 1 if the person is from the “Other” region and 0 otherwise,
and
Ube a variable that is 1 if the person is from the United States and 0 otherwise.
State a linear model for the mean of log wealth that includes age and the above 4
geographic variables. Call this the full model.
(b) (5 marks) The SSY for these data is 28.3351 and the SSE from fitting the full model
is 27.0461. Use this information to construct an ANOVA table. Your table should
include columns for Source, df, SS, MS and F, and should include rows for the model
(full model), residual and total.
(c) (2 marks) In addition to the full model, the reduced model that includes only an
intercept and age is fit. What is the null hypothesis being considered by fitting these
two models? (Refer to quantities defined in your model.)
(d) (3 marks) The SSM from the reduced model is 59.7. Calculate an F-statistic to test the hypothesis of part (c). State the degrees of freedom for the F-distribution
you would compare this statistic to. (You do not need to calculate the p-value or actually test the hypothesis.)
(e) (1 mark) Using terminology from class, name the kind of Ftest that the statistic in part (d) corresponds to.
(f) (1 mark) If the null model for these data is compared to the model that includes an
intercept and age, what are the results?

One marketing strategy used by several companies is to send personalized emails or newsletters to their customers to encourage them to buy their product. However, the opening rate of these emails is very low when the targeted customers do not feel concerned by the email. This article 1 (https://www.crop.ca/en/blog/2019/258/) will allow us to answer the following questions. Here’s an excerpt from the article :

« A quick Google search for email and newsletter opening rates leads to dozens of sources, with rates ranging from 15% to 45%. Some even claim that opening rates higher than 25% are a resounding success. But seriously, can brands afford such poor performances ? And most importantly, why are opening rates so low ?

We asked the main stakeholders : consumers. 2,189 Canadians were ques- tioned about their attitudes and behaviours regarding branded newsletters and emails. The results follow. Of all branded communications received personally, Canadians report an average opening rate of 45%. »

(a) Do the results from the survey in the article in reference allow us to affirm that on average, Canadian consumers open more than 25 % of business communications sent by email and newsletters ? Suppose that the standard deviation in the population is 20 %. Use a significance level of α = 0, 05.

(b) From what average sample rate do we reject the null hypothesis of the test carried out in (a) ?


2. (12 marks) In a study of 85 individuals, food questionnaires were used to measure the change in polyunsaturated fatty acids (PUFA) intake from the beginning to the end of the study. In addition researchers measured a genetic marker in the TNF-αgene; for this analysis, we will take the two values of this TNF-αvariable to be A- or A+. The researchers also collected information on gender, age and BMI. The primary question of interest is whether TNF-αmodifies the effect of PUFA on HDL.
 
(a) (1 mark) Let TNFa be a variable with value 1 for A+ and 0 for A- TNF-αstatus and
let gender be 1 for males and 0 for females. State a linear model for mean change in HDL that includes the predictors PUFA, TNFa, PUFA×TNFa interaction, and effects for gender, age and BMI.
(b) (1 mark) A summary of the fitted model is:
                     Estimate       Std. Error   t value     Pr(>|t|)
(Intercept)     0.037531      0.196980   0.191     0.8494
pufa              0.019143      0.009245    2.071      0.0417
TNFa              0.074433      0.034128   2.181      0.0322
gender          -0.008988      0.031793   -0.283     0.7781
age               -0.003895      0.002156   -1.806     0.0747
bmi                0.006777      0.004332    1.565      0.1217
pufa:TNFa      -0.024575      0.017844  -1.377      0.1724
Write a short sentence to describe why we would even consider removing gender from the model (i.e., why we would not just be happy with the above model from
the outset).
 
(c) (2 marks) Below is a summary of the model fit without gender. Using our 10% rule-of-thumb, is there any evidence that gender confounds the PUFA×TNFa effect?
If so why, if not why not?
                        Estimate     Std. Error      t value       Pr(>|t|)
(Intercept)       0.024467     0.190364      0.129         0.8981
pufa                0.018841     0.009130      2.064         0.0423
TNFa               0.074850     0.033897      2.208         0.0301
age                -0.003783     0.002107    -1.795         0.0765
bmi                 0.006835     0.004301     1.589          0.1161
pufa:TNFa       -0.024496     0.017737   -1.381          0.1712
 
(d) (2 marks) The researchers decided to use the model with PUFA, TNFa, PUFA×TNFa, age and BMI for inference of PUFA×TNFa interaction. What are the degrees of
freedom for the t-distribution whose critical value we need to construct confidence intervals for the interaction? What upper tail probability should you use for a 95%
confidence interval?
(e) (1 mark) The appropriate critical value is 1.96. Construct a 95% confidence interval for the interaction.
(f) (2 marks) State the null hypothesis and a two-sided alternative hypothesis for testing whether TNFa modifies the effect of PUFA on HDL.
(g) (1 mark) From the computer output, what is the p-value for this test?
(h) (2 marks) For fixed age and BMI, what is the estimated effect of a one-unit change in PUFA for a subject who is A- at TNF-α? For fixed age and BMI, what is the estimated effect of a one-unit change in PUFA for a subject who is A+ at TNF-α.

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