Lab 6: Inference for numerical dataTemplate for lab reportdownload.file("http://stat.duke.edu/courses/Spring13/sta101.001/labs/lab6.Rmd", destfile = "lab6.Rmd")Write your report, or at least run the code and create the plots, as you go so that if you get errors you canask your TA to help on the spot. Knit often to more easily determine the source of the error.North Carolina birthsIn 2004, the state of North Carolina released a large data set containing information on births recordedin this state. This data set is useful to researchers studying the relation between habits and practices ofexpectant mothers and the birth of their children. We will work with a random sample of observations fromthis data set.Exploratory analysisLoad the nc data set into our workspace.download.file("http://stat.duke.edu/courses/Spring13/sta101.001/labs/nc.RData", destfile ="nc.RData")load("nc.RData")We have observations on 13 different variables, some categorical and some numerical. The meaning of eachvariable is as follows.fage father’s age in years.mage mother’s age in years.mature maturity status of mother.weeks length of pregnancy in weeks.premie whether the birth was classified as premature (premie) or full-term.visits number of hospital visits during pregnancy.maritalwhether mother is married or not married at birth.gained weight gained by mother during pregnancy in pounds.weight weight of the baby at birth in pounds.lowbirthweight whether baby was classified as low birthweight (low) or not (not low).gender gender of the baby, female or male.habit status of the mother as a nonsmoker or a smoker.whitemom whether mom is white or not white.Exercise 1 What are the cases in this data set? How many cases are there in our sample?As a first step in an analysis, we should consider summaries of the data. This can be done using the summarycommand:This is a product of OpenIntro that is released under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported(http://creativecommons.org/licenses/by-nc-nd/3.0/). This lab was adapted for OpenIntro by Mine C¸ etinkaya-Rundel from a lab written bythe faculty and TAs of UCLA Statistics.1summary(nc)As you review the variable summaries, consider which variables are categorical and which are numerical.For numerical variables, are there outliers? If you aren’t sure or want to take a closer look at the data, makea graph.Consider the possible relationship between a mother’s smoking habit and the weight of her baby. Plottingthe data is a useful ???rst step because it helps us quickly visualize trends, identify strong associations, anddevelop research questions.Exercise 2 Make a side-by-side boxplot of habit and weight. What does the plot highlightabout the relationship between these two variables?The box plots show how the medians of the two distributions compare, and we can also compare the meansof the distributions. The following use of the by function splits the weight variable into the habit groups,then takes the mean of each using the mean function.by(nc$weight, nc$habit, mean)There is an observed difference, but is this difference statistically significant? In order to answer this questionwe will conduct a hypothesis test.InferenceExercise 3 Check if the conditions necessary for inference are satisfied. Note that you will needto obtain sample sizes to check the conditions. You can compute the group size using the sameby command above but replacing mean with length.Exercise 4 Write the hypotheses for testing if the average weights of babies born to smokingand non-smoking mothers are different.Next, we introduce a new function, inference, that we will use for conducting hypothesis tests and con-structing confidence intervals.inference(data = nc$weight, group = nc$habit, est = "mean", type = "ht", null = 0,alternative = "twosided", method = "theoretical")Let’s pause for a moment to go through the arguments of this custom function.• The first argument is data, which is the response variable that we are interested in: nc$weight.• The second argument is the grouping variable, group, which is the variable that splits the data intotwo groups, smokers and non-smokers: nc$habit.• The third argument, est, is the parameter we’re interested in: "mean" (other options are "median", or"proportion".)• Next we decide on the type of inference we want: a hypothesis test (ht) or a confidence interval ("ci").• When performing a hypothesis test, we also need to supply the null value, which in this case is 0,since the null hypothesis sets the two population means equal to each other.• The alternative hypothesis can be "less", "greater", or "twosided".2• The method of inference can be "theoretical" or "simulation" based.• Lastly, the conflevel for confidence intervals is 0.95 by default, and can be any number between 0and 1.Exercise 5 Change the type argument to "ci" construct a confidence interval for the differencebetween the weights of babies born to smoking and non-smoking mothers.By default the function reports an interval for (µnonsmoker− µsmoker), we can easily change ”this order byusing the order argument:inference(data = nc$weight, group = nc$habit, est = "mean", type = "ci",method = "theoretical", order = c("smoker","nonsmoker"))Exercise 6 Calculate a 95% confidence interval for the average length of pregnancies (weeks) andinterpret it in context. Note that since you’re doing inference on a single population parameter,there is no grouping variable, so you can omit the group variable from the function.Exercise 7 Calculate a new confidence interval for the same parameter at the 90% confidencelevel. You can change the confidence level by adding a new argument to the function: conflevel= 0.90.Exercise 8 Conduct a hypothesis test evaluating whether the average weight gained by youngermothers is different than the average weight gained by mature mothers.Exercise 9 Now, a non-inference task: Determine the age cutoff for younger and mature mothers.Use a method of your choice, and explain how your method works.Exercise 10 Pick a pair of numerical and categorical variables and come up with a researchquestion evaluating the relationship between these variables. Formulate the question in a waythat it can be answered using a hypothesis test and/or a confidence interval. Answer your ques-tion using the inference function, report the statistical results, and also provide an explanationin plain language.Exercise 11 What concepts from the textbook are