A dental research team investigating a new tooth filling material conducted a test comparing a new hardening method to the current method used by a majority of dentists. The study randomly assigned its subjects to be treated with either the new method or the current method, and measured the index of hardness for each subject after treatment. For the 62 subjects treated with the new method, the mean hardness index was 7.71 with a standard deviation of 0.73. For the 66 subjects treated with the current method, the mean hardness index was 7.48 with a standard deviation of 0.51. The standard error (SE) of the difference between the sample means is 0.112. a) Would you consider the difference in sample means to be practically significant? Why or why not? Briefly discuss the difference between statistical significance and practical significance in your response. b) Is there sufficient evidence, at the 5% significance level, to conclude that the mean index of hardness (for all patients) with the new hardening method is significantly higher than with the current method? Show your work. Conduct a complete analysis and state the conclusion in the context of the researcher’s study. c) Based on your conclusion, what type of error (Type 1 or Type 2) is possible? Why? Explain what that type of error would mean in the context of this study

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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A dental research team investigating a new tooth filling material conducted a test comparing a new hardening method to the current method used by a majority of dentists. The study randomly assigned its subjects to be treated with either the new method or the current method, and measured the index of hardness for each subject after treatment. For the 62 subjects treated with the new method, the mean hardness index was 7.71 with a standard deviation of 0.73. For the 66 subjects treated with the current method, the mean hardness index was 7.48 with a standard deviation of 0.51. The standard error (SE) of the difference between the sample means is 0.112.

a) Would you consider the difference in sample means to be practically significant? Why or why not? Briefly discuss the difference between statistical significance and practical significance in your response.

b) Is there sufficient evidence, at the 5% significance level, to conclude that the mean index of hardness (for all patients) with the new hardening method is significantly higher than with the current method? Show your work. Conduct a complete analysis and state the conclusion in the context of the researcher’s study.

c) Based on your conclusion, what type of error (Type 1 or Type 2) is possible? Why? Explain what that type of error would mean in the context of this study

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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