A paint manufacturer wishes to compare the drying times of two different types of paint. Independent random samples of 11 cans of type A and 9 cans of type B were selected and applied to similar surfaces. The drying times (in hours) were recorded. The summary statistics are given in the image below. Use a 0.01 significance level to test the claim that the mean drying time for paint type A is longer than the mean drying time for paint type B. Find each:a) Set up (including Null and Alternative Hypothesis) b) Test Statisticc) P-value d) Conclusion (reject or accept H0). ### Comparative Analysis of Two TypesThis table represents a comparative analysis between two distinct types, labeled Type A and Type B. The comparison includes the mean ($\bar{x}$), standard deviation ($s$), and sample size ($n$) for each type.#### Type A:- **Mean ($\bar{x}_1$)**: 76.9 - The average value for Type A is 76.9.- **Standard Deviation ($s_1$)**: 4.5 - This indicates the variability or dispersion of the data points around the mean for Type A is 4.5.- **Sample Size ($n_1$)**: 11 - The number of observations or data points collected for Type A is 11.#### Type B:- **Mean ($\bar{x}_2$)**: 66.3 - The average value for Type B is 66.3.- **Standard Deviation ($s_2$)**: 5.1 - This signifies the variability or dispersion of the data points around the mean for Type B is 5.1.- **Sample Size ($n_2$)**: 9 - The number of observations or data points collected for Type B is 9.This analysis provides a summary of the central tendencies and variability within both groups, which serves as a fundamental basis for further statistical comparison, such as t-tests or variance analyses, to determine if the observed differences are statistically significant.

Answered: A paint manufacturer wishes to compare…

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 paint manufacturer wishes to compare the drying times of two different types of paint. Independent random samples of 11 cans of type A and 9 cans of type B were selected and applied to similar surfaces. The drying times (in hours) were recorded. The summary statistics are given in the image below. Use a 0.01 significance level to test the claim that the mean drying time for paint type A is longer than the mean drying time for paint type B. Find each:

a) Set up (including Null and Alternative Hypothesis)

b) Test Statistic

c) P-value

d) Conclusion (reject or accept H₀).

### Comparative Analysis of Two Types

This table represents a comparative analysis between two distinct types, labeled Type A and Type B. The comparison includes the mean ($\bar{x}$), standard deviation ($s$), and sample size ($n$) for each type.

#### Type A:
- **Mean ($\bar{x}_1$)**: 76.9
- The average value for Type A is 76.9.
- **Standard Deviation ($s_1$)**: 4.5
- This indicates the variability or dispersion of the data points around the mean for Type A is 4.5.
- **Sample Size ($n_1$)**: 11
- The number of observations or data points collected for Type A is 11.

#### Type B:
- **Mean ($\bar{x}_2$)**: 66.3
- The average value for Type B is 66.3.
- **Standard Deviation ($s_2$)**: 5.1
- This signifies the variability or dispersion of the data points around the mean for Type B is 5.1.
- **Sample Size ($n_2$)**: 9
- The number of observations or data points collected for Type B is 9.

This analysis provides a summary of the central tendencies and variability within both groups, which serves as a fundamental basis for further statistical comparison, such as t-tests or variance analyses, to determine if the observed differences are statistically significant.

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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