Answer :
To find the test statistic and the p-value for the given hypothesis test, we'll follow these steps:
### Step 1: Understand the Setup
We are conducting a hypothesis test for the population mean [tex]\(\mu\)[/tex] with the following hypotheses:
- Null Hypothesis [tex]\(H_0: \mu = 75.2\)[/tex]
- Alternative Hypothesis [tex]\(H_a: \mu \neq 75.2\)[/tex]
The significance level is [tex]\(\alpha = 0.005\)[/tex].
### Step 2: Collect and Analyze the Sample Data
You have a sample data set with 55 observations. Given that the population standard deviation is unknown and the sample size is relatively large but not enormous, we will use the t-distribution.
### Step 3: Calculate the Sample Mean and Sample Standard Deviation
1. Sample Mean ([tex]\(\bar{x}\)[/tex]): Add all the sample values together and divide by the number of observations to find the sample mean.
2. Sample Standard Deviation (s): Use the formula for standard deviation with Bessel's correction (using [tex]\(n-1\)[/tex] as the denominator).
### Step 4: Calculate the Test Statistic
The test statistic for the mean when the population standard deviation is unknown is a t-statistic, given by the formula:
[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 under the null hypothesis ([tex]\(75.2\)[/tex])
- [tex]\(s\)[/tex] is the sample standard deviation
- [tex]\(n\)[/tex] is the sample size
After performing the calculation, the test statistic is found to be:
[tex]\[
t \approx -4.794
\][/tex]
### Step 5: Calculate the p-value
Given that the alternative hypothesis is two-tailed ([tex]\(H_a: \mu \neq 75.2\)[/tex]), the p-value is calculated by finding the probability that a t-distribution with [tex]\(n-1\)[/tex] degrees of freedom is more extreme than the observed test statistic in either tail.
Using statistical software or a t-distribution table, you find the p-value to be:
[tex]\[
\text{p-value} \approx 0.0000
\][/tex]
### Conclusion
Finally, with a test statistic of approximately [tex]\(-4.794\)[/tex] and a p-value of approximately [tex]\(0.0000\)[/tex], you compare the p-value to the significance level [tex]\(\alpha = 0.005\)[/tex]. Since the p-value is smaller than [tex]\(\alpha\)[/tex], you reject the null hypothesis [tex]\(H_0\)[/tex]. This suggests there is significant evidence to support the claim that the mean is different from 75.2.
### Step 1: Understand the Setup
We are conducting a hypothesis test for the population mean [tex]\(\mu\)[/tex] with the following hypotheses:
- Null Hypothesis [tex]\(H_0: \mu = 75.2\)[/tex]
- Alternative Hypothesis [tex]\(H_a: \mu \neq 75.2\)[/tex]
The significance level is [tex]\(\alpha = 0.005\)[/tex].
### Step 2: Collect and Analyze the Sample Data
You have a sample data set with 55 observations. Given that the population standard deviation is unknown and the sample size is relatively large but not enormous, we will use the t-distribution.
### Step 3: Calculate the Sample Mean and Sample Standard Deviation
1. Sample Mean ([tex]\(\bar{x}\)[/tex]): Add all the sample values together and divide by the number of observations to find the sample mean.
2. Sample Standard Deviation (s): Use the formula for standard deviation with Bessel's correction (using [tex]\(n-1\)[/tex] as the denominator).
### Step 4: Calculate the Test Statistic
The test statistic for the mean when the population standard deviation is unknown is a t-statistic, given by the formula:
[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 under the null hypothesis ([tex]\(75.2\)[/tex])
- [tex]\(s\)[/tex] is the sample standard deviation
- [tex]\(n\)[/tex] is the sample size
After performing the calculation, the test statistic is found to be:
[tex]\[
t \approx -4.794
\][/tex]
### Step 5: Calculate the p-value
Given that the alternative hypothesis is two-tailed ([tex]\(H_a: \mu \neq 75.2\)[/tex]), the p-value is calculated by finding the probability that a t-distribution with [tex]\(n-1\)[/tex] degrees of freedom is more extreme than the observed test statistic in either tail.
Using statistical software or a t-distribution table, you find the p-value to be:
[tex]\[
\text{p-value} \approx 0.0000
\][/tex]
### Conclusion
Finally, with a test statistic of approximately [tex]\(-4.794\)[/tex] and a p-value of approximately [tex]\(0.0000\)[/tex], you compare the p-value to the significance level [tex]\(\alpha = 0.005\)[/tex]. Since the p-value is smaller than [tex]\(\alpha\)[/tex], you reject the null hypothesis [tex]\(H_0\)[/tex]. This suggests there is significant evidence to support the claim that the mean is different from 75.2.
