Suppose the National Transportation Safety Board (NTSB) wants to examine the safety of compact cars, midsize cars, and full-size cars. It collects a sample of three for each of the treatments (cars types). Using the hypothetical data provided below, run your analyses in R to test whether the mean pressure applied to the driver’s head during a crash test is equal for each types of car (higher pressure = less safe). Compact CarMidsize CarFull-Size Car643469484655427456702525402 What kind of statistical analysis do you need to use and why? State the null and alternative hypotheses.Check your assumptions. Include screenshots of your R output and explain why each of them are or are not met. Using R, run your analysis and include a screenshot. Using R, run post-hoc tests and include a screenshot. What do you conclude about the null hypothesis? What does each of your p-values tell you? Are there differences and in which comparisons? Explain what your p-values are telling you regarding the safety of the cars.

1. ANOVA: The analysis of variance test is used to compare the means of more than two groups. In one way ANOVA model, only one factor is studied, that is only the effect on one factor is studied simultaneously. In this model there would be one dependent variable and one independent variable (with more than 2 levels). Explanation: In the given situation, the researcher testing whether the mean pressure applied to the driver’s head during a crash test is equal for each types of car or not. The suitable statistical test is 'one way ANOVA'. 2. Null Hypothesis: H0: the mean pressure applied to the driver’s head during a crash test is equal for each types of cars. Alternative Hypothesis: H1: At least two types of cars have different mean pressures. 3. Assumptions for ANOVA: Samples are taken at random and are independent of each other. Samples are taken from a normally distributed population. The variances of probability distributions are equal. Homogeneity of Variance: Null Hypothesis: H0: The three car types having same variance. Alternative Hypothesis: H1: At least two car types having different variances. R syntax: From the output, the p value is 0.8485. Decision Rule: If p-value ≤ α, then reject the null hypothesis. If p-value > α, then do not reject the null hypothesis. Conclusion: Let the level of significance is α=0.05. Here, the p-value is greater than the level of significance. From the rejection rule, failed to reject the null hypothesis. The three car types having same variance. Testing of Normality: Null Hypothesis: H0: The 'Compact' data follows normal distribution Alternative Hypothesis: H1: The 'Compact' data does not follow normal distribution From the output, the p value is 0.3698. Conclusion: Let the level of significance is α=0.05. Here, the p-value is greater than the level of significance. From the rejection rule, failed to reject the null hypothesis. The 'Compact' data follows normal distribution. Null Hypothesis: H0: The 'Midsize' data follows normal distribution Alternative Hypothesis: H1: The 'Midsize' data does not follow normal distribution From the output, the p value is 0.8428. Conclusion: Let the level of significance is α=0.05. Here, the p-value is greater than the level of significance. From the rejection rule, failed to reject the null hypothesis. The 'Midsize' data follows normal distribution. Null Hypothesis: H0: The 'Full-size' data follows normal distribution Alternative Hypothesis: H1: The 'Full-size' data does not follow normal distribution From the output, the p value is 0.6542 Conclusion: Let the level of significance is α=0.05. Here, the p-value is greater than the level of significance. From the rejection rule, failed to reject the null hypothesis. The 'Full-size' data follows normal distribution. Hence, the assumptions for one way ANOVA are satisfied. 4. ANOVA: 5. Multiple comparison tests: 6) From the part 4, the p value is 0.00121. Conclusion: Let the level of significance is α=0.05. Here, the p-value is less than the level of significance. From the rejection rule, reject the null hypothesis. 7) From the part 5, Pair p value Conclusion Fullsize-compact 0.0015 p <0.05 Significant pair Midsize-compact 0.002997 p<0.05 Significant pair Midsize-Fullsize 0.727875 p>0.05 Not significant Therefore, There is exists a significant difference between the means of Full-size and compact type cars. There is exists a significant difference between the means of Mid-size and compact type cars. There is exists no significant difference between the means of Mid-size and Full-size type cars. 8. From the part 6, the null hypothesis is rejected. That is, the mean pressure applied to the driver’s head during a crash test is not equal for each types of car.

Answered: Suppose the National Transportation…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

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Suppose the National Transportation Safety Board (NTSB) wants to examine the safety of compact cars, midsize cars, and full-size cars. It collects a sample of three for each of the treatments (cars types). Using the hypothetical data provided below, run your analyses in R to test whether the mean pressure applied to the driver’s head during a crash test is equal for each types of car (higher pressure = less safe).

Compact Car	Midsize Car	Full-Size Car
643	469	484
655	427	456
702	525	402

What kind of statistical analysis do you need to use and why?
State the null and alternative hypotheses.
Check your assumptions. Include screenshots of your R output and explain why each of them are or are not met.
Using R, run your analysis and include a screenshot.
Using R, run post-hoc tests and include a screenshot.
What do you conclude about the null hypothesis?
What does each of your p-values tell you? Are there differences and in which comparisons?
Explain what your p-values are telling you regarding the safety of the cars.

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

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