Please justify your answer to each question. This justification can involve hand calculation or providing relevant interpretation of output from jamovi and/or statstar.io. If a question requires a calculation, please show your working. If a question requires output from statstar.io, please provide and refer to this output accordingly. Do not simply copy and paste jamovi or statstar.io output, but provide concise interpretation of this output where appropriate. Statistics is about communication as much as it is about calculations. Please provide your answers in a sentence to communicate your response.
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It has been suggested for over 2,000 years that warmer water will freeze faster than colder water. Even as far back as Aristotle this was thought with the Greek philosopher stating:
“The fact that the water has previously been warmed contributes to its freezing quickly: for so it cools sooner. Hence many people, when they want to cool water quickly, begin by putting it in the sun.”
A high school science teacher asked their class to help take part in an experiment to test this idea. In order to conduct the testing, they randomly allocated each student to one of two groups. The first group would test the time it takes for water to freeze when starting with hot water. The second group would time how long it takes for water to freeze when starting with cold water.
Students were each supplied with a wireless infrared thermometer in order to set the initial tem- peratures of the water to a set temperature depending on what group they were allocated to. The thermometers could also be used to measure when the water is frozen without the need to open the freezer door in their home kitchen.
Data collected for the time taken for the water to freeze in minutes is shown below:
Hot Cold
285 307
304 307
303 295
294 300
288 310
292 324
313 312
299 326
316 315
323 305
(a) [2 marks] Name an instrument or something else related to the experiment that the teacher should clearly prescribe or describe to the students. The goal of this would be to reduce the amount of variation in the data.
(b) [1 mark] Is the data paired or not? Explain why.
(c) [3 marks] Enter the data into jamovi and produce a graph to investigate the relationship between the time it takes water to freeze and the initial temperature of the water. Be sure to compare the two distributions. Hint: Paired and independent data needs to be set out differently in jamovi.
(d) [6 marks] Carry out an appropriate hypothesis test to investigate if there is a difference in the population mean times for water to freeze based on initial temperature. Make sure to report the hypotheses tested, test statistic, null distribution, p-value and conclusion in plain language.
(e) [3 marks] Produce a 95% confidence interval for the difference in time it takes for water to freeze with either hot or cold initial conditions. Give the confidence interval in a sentence. Does this confidence interval support your decision in part (d)?
(f) [4 marks] State and check whether the assumptions of this test have been met.
(g) [2 marks] If we reduce the amount of variation in data, we are more likely to be able to identify differences. With that in mind, provide one suggestion to improve the experiment?
The data file sample cars.omv contains information from the United States of America for 18 randomly sampled motor vehicles. The data was collected to investigate whether there are differences in fuel economy for motor vehicles with differing drive types.
The two variables of interest here are MPG City and DriveTrain. The MPG variable is a measure of distance travelled per gallon of fuel used when driving in a city environment. The larger this number the more fuel efficient the vehicle is. The drive train variable indicates what wheels of the vehicle are mechanically driven from the engine.
It is expected that front wheel drives have the best fuel economy, followed by rear wheel drives and then all wheel drives. This is due to the mechanical losses in the drive train.
For the following, use a significance level of 5%.
(a) [2 marks] Plot the data and provide a preliminary assessment of whether there appears to be differences between fuel efficiency based on drive type.
(b) [6 marks] Carry out an appropriate hypothesis test to for differences in fuel efficiency for the different drive types. Make sure to report the hypotheses tested, test statistic, null distribution, p-value and conclusion in plain language.
(c) [3 marks] If appropriate, carry out post-hoc tests and report the results of each comparison making sure you justify your conclusions. If post-hoc testing is not appropriate, explain the purpose of post-hoc tests and why they are not appropriate in this example.
(d) [3 marks] Assess the assumptions of the test you performed in part (b).
In the previous question, researchers randomly collected data on motor vehicles from the United States of America. Another insight they are hoping to receive from the data is if there is a relationship between weight and the fuel economy of a vehicle.
The variables they are now interested in is are Weight which is measured in pounds (lb) and MPG Highway. The MPG variable is a measure of distance travelled per gallon of fuel used when driving on a highway.
The data appear in the file sample cars.omv.
(a) [3 marks] A scatterplot is a useful graphical tool for investigating possible relationships between two numerical variables. Produce a scatterplot using the data provided ensuring that you place the appropriate variables on the x-axis and y-axis. Describe the relationship between the two variables.
(b) [3 marks] Provide the equation for the linear regression model of highway MPG and weight. Interpret the numbers (b0 and b1) in the regression equation.
(c) [1 mark] Use the regression equation to predict the highway MPG of a vehicle with a weight of 3,000lb.
(d) [1 mark] How much of the variation in highway MPG is accounted for in the linear regression with weight as a predictor?
(e) [6 marks] Test at the 5% significance level if there is a relationship between critical highway MPG and weight. Be sure to include the following in your answer:
the null and alternative hypotheses
the test statistic
the null distribution
the p-value
your conclusion
(f) [2 marks] Provide a 95% confidence interval for the population slope parameter and interpret this interval.
(g) [4 marks] List the assumptions of your analysis. To what degree are these assumptions satisfied? Include output from jamovi where appropriate.
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