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Answer :
To determine the test statistic for this sample with the given hypothesis, we'll go through the process step-by-step.
### Step 1: Collect the Sample Data
The sample consists of the following data points:
61.6, 113.7, 64.4, 75.1, 75.5, 52, 60.9, 82.9, 39.7, 64.4, 87.4, 88.5, 64.4, 85.1, 63.3, 65, 46.8, 100.2, 60.2, 95.4, 82.4, 89, 104.6, 68.4, 79.1, 84.2, 53.3, 60.9, 72.7, 90.1, 44.1, 53.3, 66.5, 65, 95.4, 109.4, 84.6, 58, 84.6, 78.7, 55.4, 84.2, 62.8, 79.9, 90.1, 99.1, 86, 99.1, 59.5, 91.3, 52, 79.1, 61.6, 88, 44.1
### Step 2: Calculate the Sample Mean
First, we calculate the sample mean [tex]\(\bar{x}\)[/tex]:
[tex]\[
\bar{x} = \frac{\text{sum of all sample values}}{\text{number of samples}}
\][/tex]
The sample mean is 74.6.
### Step 3: Calculate the Sample Standard Deviation
The sample standard deviation [tex]\(s\)[/tex] is calculated using the formula:
[tex]\[
s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}}
\][/tex]
where [tex]\(x_i\)[/tex] are the sample values and [tex]\(n\)[/tex] is the number of samples.
The sample standard deviation is 17.821 (rounded to three decimal places).
### Step 4: Calculate the Test Statistic
The test statistic is calculated using the formula for the t-statistic:
[tex]\[
t = \frac{\bar{x} - \mu}{s / \sqrt{n}}
\][/tex]
where:
- [tex]\(\bar{x}\)[/tex] is the sample mean,
- [tex]\(\mu\)[/tex] is the population mean (83.8 in this case),
- [tex]\(s\)[/tex] is the sample standard deviation,
- [tex]\(n\)[/tex] is the sample size.
Given the calculations:
- [tex]\(\bar{x} = 74.6\)[/tex]
- [tex]\(\mu = 83.8\)[/tex]
- [tex]\(s = 17.821\)[/tex]
- [tex]\(n = 55\)[/tex]
Substituting these into the formula gives us the test statistic:
[tex]\[
t = \frac{74.6 - 83.8}{17.821 / \sqrt{55}} = -3.829
\][/tex]
Thus, the test statistic is [tex]\(-3.829\)[/tex] (rounded to three decimal places).
### Step 1: Collect the Sample Data
The sample consists of the following data points:
61.6, 113.7, 64.4, 75.1, 75.5, 52, 60.9, 82.9, 39.7, 64.4, 87.4, 88.5, 64.4, 85.1, 63.3, 65, 46.8, 100.2, 60.2, 95.4, 82.4, 89, 104.6, 68.4, 79.1, 84.2, 53.3, 60.9, 72.7, 90.1, 44.1, 53.3, 66.5, 65, 95.4, 109.4, 84.6, 58, 84.6, 78.7, 55.4, 84.2, 62.8, 79.9, 90.1, 99.1, 86, 99.1, 59.5, 91.3, 52, 79.1, 61.6, 88, 44.1
### Step 2: Calculate the Sample Mean
First, we calculate the sample mean [tex]\(\bar{x}\)[/tex]:
[tex]\[
\bar{x} = \frac{\text{sum of all sample values}}{\text{number of samples}}
\][/tex]
The sample mean is 74.6.
### Step 3: Calculate the Sample Standard Deviation
The sample standard deviation [tex]\(s\)[/tex] is calculated using the formula:
[tex]\[
s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}}
\][/tex]
where [tex]\(x_i\)[/tex] are the sample values and [tex]\(n\)[/tex] is the number of samples.
The sample standard deviation is 17.821 (rounded to three decimal places).
### Step 4: Calculate the Test Statistic
The test statistic is calculated using the formula for the t-statistic:
[tex]\[
t = \frac{\bar{x} - \mu}{s / \sqrt{n}}
\][/tex]
where:
- [tex]\(\bar{x}\)[/tex] is the sample mean,
- [tex]\(\mu\)[/tex] is the population mean (83.8 in this case),
- [tex]\(s\)[/tex] is the sample standard deviation,
- [tex]\(n\)[/tex] is the sample size.
Given the calculations:
- [tex]\(\bar{x} = 74.6\)[/tex]
- [tex]\(\mu = 83.8\)[/tex]
- [tex]\(s = 17.821\)[/tex]
- [tex]\(n = 55\)[/tex]
Substituting these into the formula gives us the test statistic:
[tex]\[
t = \frac{74.6 - 83.8}{17.821 / \sqrt{55}} = -3.829
\][/tex]
Thus, the test statistic is [tex]\(-3.829\)[/tex] (rounded to three decimal places).
Thank you for reading the article You wish to test the following claim tex H a tex at a significance level of tex alpha 0 01 tex tex begin array l. We hope the information provided is useful and helps you understand this topic better. Feel free to explore more helpful content on our website!
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Rewritten by : Jeany
