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. ### Statistical Comparison of Type A and Type BThe table below presents a comparison of two types, namely Type A and Type B, based on their respective means, standard deviations, and sample sizes.| | Type A | Type B ||----------|----------|----------|| Mean ($\bar{x}$) | 76.9 | 66.3 || Standard Deviation ($s$) | 4.5 | 5.1 || Sample Size ($n$)| 11 | 9 |#### Detailed Explanation:- **Mean ($\bar{x}$)**: This represents the average value for each type. - Type A: The mean value is 76.9. - Type B: The mean value is 66.3.- **Standard Deviation ($s$)**: This measures the amount of variation or dispersion from the mean. - Type A: The standard deviation is 4.5, indicating less variability around the mean compared to Type B. - Type B: The standard deviation is 5.1, indicating more variability around the mean compared to Type A.- **Sample Size ($n$)**: This denotes the number of observations in each sample. - Type A: The sample size is 11. - Type B: The sample size is 9.The data provided can be used for various statistical analyses such as hypothesis testing, confidence interval calculations, or comparative studies to determine significant differences between Type A and Type B. **Linear Correlation**### Critical Values and Statistical Significance*If $ |r| > CV $, then correlation is statistically significant.* n Critical Value (CV) 4 .950 5 .878 6 .811 7 .754 8 .707 9 .666 10 .632 11 .602 12 .576 13 .553 14 .532 15 .514 This table lists the critical values (CV) for a linear correlation when considering different sample sizes (n). When the absolute value of the correlation coefficient $ |r| $ exceeds the critical value (CV) corresponding to the sample size, the correlation is deemed statistically significant.

A t-test is a type of inferential statistic used to determine if there is a significant difference…

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.

### Statistical Comparison of Type A and Type B

The table below presents a comparison of two types, namely Type A and Type B, based on their respective means, standard deviations, and sample sizes.

| | Type A | Type B |
|----------|----------|----------|
| Mean ($\bar{x}$) | 76.9 | 66.3 |
| Standard Deviation ($s$) | 4.5 | 5.1 |
| Sample Size ($n$)| 11 | 9 |

#### Detailed Explanation:

- **Mean ($\bar{x}$)**: This represents the average value for each type.
- Type A: The mean value is 76.9.
- Type B: The mean value is 66.3.

- **Standard Deviation ($s$)**: This measures the amount of variation or dispersion from the mean.
- Type A: The standard deviation is 4.5, indicating less variability around the mean compared to Type B.
- Type B: The standard deviation is 5.1, indicating more variability around the mean compared to Type A.

- **Sample Size ($n$)**: This denotes the number of observations in each sample.
- Type A: The sample size is 11.
- Type B: The sample size is 9.

The data provided can be used for various statistical analyses such as hypothesis testing, confidence interval calculations, or comparative studies to determine significant differences between Type A and Type B.

**Linear Correlation**

### Critical Values and Statistical Significance
*If $ |r| > CV $, then correlation is statistically significant.*

<table>
<tr>
<th>n</th>
<th>Critical Value (CV)</th>
</tr>
<tr>
<td>4</td>
<td>.950</td>
</tr>
<tr>
<td>5</td>
<td>.878</td>
</tr>
<tr>
<td>6</td>
<td>.811</td>
</tr>
<tr>
<td>7</td>
<td>.754</td>
</tr>
<tr>
<td>8</td>
<td>.707</td>
</tr>
<tr>
<td>9</td>
<td>.666</td>
</tr>
<tr>
<td>10</td>
<td>.632</td>
</tr>
<tr>
<td>11</td>
<td>.602</td>
</tr>
<tr>
<td>12</td>
<td>.576</td>
</tr>
<tr>
<td>13</td>
<td>.553</td>
</tr>
<tr>
<td>14</td>
<td>.532</td>
</tr>
<tr>
<td>15</td>
<td>.514</td>
</tr>
</table>

This table lists the critical values (CV) for a linear correlation when considering different sample sizes (n). When the absolute value of the correlation coefficient $ |r| $ exceeds the critical value (CV) corresponding to the sample size, the correlation is deemed 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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