MATH 1068 Statistical Methods Assignment

Question 1 (18 Marks)

Microwave Radiation:

The maximum acceptable level for exposure to microwave radiation is an average of 10 micro watts per square centimetre. It is feared that large television transmitters may be polluting the air nearby by pushing the level of microwave radiation above the safe limit.

1. (2 marks) The following randomly selected sample of 12 observations were taken allocations near the transmitter and measured in the number of micro watts per square centimetre.
   9 |11 |14 |10 |10 |12 |13 |8 |12 |14 |6 |18
   Use MINITAB to calculate the sample mean, sample standard deviation and test for Normality.
2. (8 marks) Statistically test at a 10% level of significance whether the microwave radiation is above the safe limit. Include the set-up of the null and alternative hypotheses, a diagram and conclusion in your answer. All calculations should be done without using MINITAB.
3. (6 marks) Another randomly selected sample of 10 observations were recorded for another transmitter given below. Conduct a statistical test at a 5% level of significance to determine whether there is a difference in radiation between the transmitters.
   11 |10 |12 |11 |18 |14 |10 |10 |15 |11
4. (2 marks) What is the meaning of a Type I Error? Does this mean there is an ? probability that a Type I Error was made?

Question 2 (37 Marks)

Concentration Data:

A study was conducted across three cities to investigate mineral concentrations in household tap water and susceptibility to illness. The dataset concentrations.MTW displays the mineral concentrations and the average number of people suffering from an illness in a suburb. The data is across a year and is the measure of both Calcium and Magnesium Mineral Concentrations (in parts per million (ppm)) in household drinking water for a total of 79 large suburbs in three cities. If there is a higher Mineral Concentration, then this physically translates to the water being more cohesive and believed to be healthy.

1. (5 marks) Use Minitab to produce a scatterplot and compute the correlation coefficient between Illness and Mineral Concentration. Please provide an interpretation of the association between Illness and Mineral Concentration.
2. (7 marks) The Australian Water Board claims for health reasons that the Mineral Concentration in water should be above 60 parts per million. At the 5% level of significance, test whether the average Mineral Concentration is above 60 ppm.
3. (10 marks) Check the assumptions for ANOVA and test at the 5% significance level to determine whether there is a difference in Mineral Concentration between the different cities. Include the post-hoc Tukey test to identify which cities are different, if appropriate. Include your MINITAB output for full marks.
4. (3 marks) The National Health Board is interested to know if the illness numbers are equal to or exceed 1500 in a suburb and the comparison across the three cities. Construct a contingency table to explain this from the data collected. The Illness Level variable can be decomposed into the factors; greater than or equal to 1500 and less than 1500.
5. (5 marks) Conduct an appropriate hypothesis test at a 5% level of significance to determine whether the Illness level is related to the cities.
6. (3 marks) What conclusions can be drawn from the residuals in the table? State all the significant residuals in the answer.
7. (4 marks) By examining the patterns in the data across the different cities, make a recommendation about which cities have higher mineral concentrations and the relationship between cities and illness. Summarise the results of your analyses from parts (a), (c), (e) and (f) (3-4 sentences).

Question 3 (35 Marks)

St Kilda Housing – Price Prediction

The dataset StKildaHousing.csv contains house data on houses (h) and units (u). A real estate agency wants to create a model to predict the Price ($) in St Kilda. The following questions below will guide you through creating a predictive model based on the data provided.

For full marks, ensure that appropriate axis labels and meaningful titles are included with all your graphical displays for this question.

Hint: Consider sorting your data and only using non-missing entries.

1. (5 marks) Use Minitab to compute the correlation coefficient between Price ($) and the other quantitative variables. Produce a scatterplot of each meaningful relationship to describe the association between the two variables. Please provide an interpretation of the observed relationship with these variables for full marks.
2. (6 marks) Run a linear regression model in MINITAB with the highest explained variable(s) to predict Price ($). Explain whether the requirements for a linear regression model are satisfied. The requirements to verify and explain are Linearity, Independence, Normality and Population standard deviations. For full marks attach your MINITAB output.
3. (5 marks) What is the value of the intercept? Is the intercept meaningful, if it is not meaningful then explain why and produce a zero-intercept model?
4. (2 marks) What is the value of the slope? What does the slope measure in this scenario?
5. (3 marks) What is the value of the coefficient of determination? What precisely does it measure in this example?
6. (5 marks) The regression model from the MINITAB output predicts the Price given the your chosen input variable. A property has approximately 100sqm of Building Area, 3 rooms and an approximate Land size of 200sqm. Use the regression model to predict the Price ($) on average of this property? Is this prediction accurate? Give reasons.
7. (4 marks) Based on your answer from part (f), use MINITAB to calculate the 95% prediction interval and the margin of error for the predicted value. State the formula for the 95% prediction interval and attach your MINITAB output for full marks.
8. (5 marks) Investigate including more variables as the predictor variables and determine if there is a more accurate linear model. Give reasons and briefly explain. By using the model, calculate the Price ($) of the property in part (f).