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Answer :
To solve the given hypothesis testing problem for the pulse rates, we'll follow these steps:
1. State the Hypotheses:
- The null hypothesis ([tex]\(H_0\)[/tex]) is a statement of no effect or no difference. Here, it states that the population mean pulse rate ([tex]\(\mu\)[/tex]) is equal to 69 bpm:
[tex]\[
H_0: \mu = 69 \text{ bpm}
\][/tex]
- The alternative hypothesis ([tex]\(H_1\)[/tex]) is what we want to test. In this case, we are testing if the mean is less than 69 bpm:
[tex]\[
H_1: \mu < 69 \text{ bpm}
\][/tex]
- So, the correct choice for the hypotheses is B:
[tex]\[
H_0: \mu = 69 \text{ bpm} \quad \text{and} \quad H_1: \mu < 69 \text{ bpm}
\][/tex]
2. Calculate the Test Statistic:
Since we are testing a mean with an unknown population standard deviation, we will use the t-test. Here is how you calculate the test statistic (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 stated in the null hypothesis (69 bpm).
- [tex]\(s\)[/tex] is the sample standard deviation.
- [tex]\(n\)[/tex] is the sample size.
3. Determine the Values Needed:
- You will first need to calculate the sample mean [tex]\(\bar{x}\)[/tex] and the sample standard deviation [tex]\(s\)[/tex] from your data set.
- Example:
- If the data set is [70, 68, 71, 67, 65] (just an example),
- Sample Mean ([tex]\(\bar{x}\)[/tex]) = (70 + 68 + 71 + 67 + 65) / 5 = 68.2 bpm.
- Sample Standard Deviation ([tex]\(s\)[/tex]) can be calculated using the formula for standard deviation.
4. Substitute the Values in the Formula:
- Assume:
- [tex]\(\bar{x} = 68.2\)[/tex]
- [tex]\(s = 2.4\)[/tex] (this is just an assumed value; you need to calculate based on actual data)
- [tex]\(n = 5\)[/tex]
- Then:
[tex]\[
t = \frac{68.2 - 69}{2.4/\sqrt{5}}
\][/tex]
5. Calculate the t-value:
- After calculating, round the t-statistic to two decimal places to fulfill the requirement of the question.
6. Make Your Decision:
- Compare the calculated t-statistic with the critical value from the t-distribution table for the degrees of freedom ([tex]\(n-1\)[/tex]) at the 0.01 significance level.
- If the calculated t-statistic is less than the negative of the critical value, reject the null hypothesis ([tex]\(H_0\)[/tex]).
Remember, the particular numeric values depend on your original dataset, which you need to analyze.
1. State the Hypotheses:
- The null hypothesis ([tex]\(H_0\)[/tex]) is a statement of no effect or no difference. Here, it states that the population mean pulse rate ([tex]\(\mu\)[/tex]) is equal to 69 bpm:
[tex]\[
H_0: \mu = 69 \text{ bpm}
\][/tex]
- The alternative hypothesis ([tex]\(H_1\)[/tex]) is what we want to test. In this case, we are testing if the mean is less than 69 bpm:
[tex]\[
H_1: \mu < 69 \text{ bpm}
\][/tex]
- So, the correct choice for the hypotheses is B:
[tex]\[
H_0: \mu = 69 \text{ bpm} \quad \text{and} \quad H_1: \mu < 69 \text{ bpm}
\][/tex]
2. Calculate the Test Statistic:
Since we are testing a mean with an unknown population standard deviation, we will use the t-test. Here is how you calculate the test statistic (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 stated in the null hypothesis (69 bpm).
- [tex]\(s\)[/tex] is the sample standard deviation.
- [tex]\(n\)[/tex] is the sample size.
3. Determine the Values Needed:
- You will first need to calculate the sample mean [tex]\(\bar{x}\)[/tex] and the sample standard deviation [tex]\(s\)[/tex] from your data set.
- Example:
- If the data set is [70, 68, 71, 67, 65] (just an example),
- Sample Mean ([tex]\(\bar{x}\)[/tex]) = (70 + 68 + 71 + 67 + 65) / 5 = 68.2 bpm.
- Sample Standard Deviation ([tex]\(s\)[/tex]) can be calculated using the formula for standard deviation.
4. Substitute the Values in the Formula:
- Assume:
- [tex]\(\bar{x} = 68.2\)[/tex]
- [tex]\(s = 2.4\)[/tex] (this is just an assumed value; you need to calculate based on actual data)
- [tex]\(n = 5\)[/tex]
- Then:
[tex]\[
t = \frac{68.2 - 69}{2.4/\sqrt{5}}
\][/tex]
5. Calculate the t-value:
- After calculating, round the t-statistic to two decimal places to fulfill the requirement of the question.
6. Make Your Decision:
- Compare the calculated t-statistic with the critical value from the t-distribution table for the degrees of freedom ([tex]\(n-1\)[/tex]) at the 0.01 significance level.
- If the calculated t-statistic is less than the negative of the critical value, reject the null hypothesis ([tex]\(H_0\)[/tex]).
Remember, the particular numeric values depend on your original dataset, which you need to analyze.
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