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Sagot :
To tackle this problem, let's go through the steps to test the hypothesis using the traditional method. The goal is to test the claim that the sample pulse rates come from a population with a mean less than 72 beats per minute, given the sample data and the known population standard deviation.
### Step 1: State the Hypotheses
First, we need to set up our null and alternative hypotheses:
- [tex]\( H_0 \)[/tex]: The population mean pulse rate is equal to 72 beats per minute ([tex]\( \mu = 72 \)[/tex]).
- [tex]\( H_1 \)[/tex]: The population mean pulse rate is less than 72 beats per minute ([tex]\( \mu < 72 \)[/tex]).
Hence, the correct hypothesis from the provided choices is:
[tex]\[ H_0: \mu = 72 \text{ beats per minute} \][/tex]
[tex]\[ H_1: \mu < 72 \text{ beats per minute} \][/tex]
### Step 2: Gather Sample Data and Statistics
From the sample data provided, we have resting pulse rates:
[tex]\[ [56, 59, 69, 84, 74, 64, 69, 70, 66, 68, 59, 71, 76, 63]. \][/tex]
Given:
- Population standard deviation ([tex]\( \sigma \)[/tex]): 6.6 beats per minute
- Significance level ([tex]\( \alpha \)[/tex]): 0.05
For these data, we know:
- Sample size ([tex]\( n \)[/tex]): 14
- Sample mean ([tex]\( \bar{x} \)[/tex]): 67.71 beats per minute (approx.)
### Step 3: Calculate the Test Statistic
We calculate the standard error of the mean (SEM) using the formula:
[tex]\[ \text{SEM} = \frac{\sigma}{\sqrt{n}} = \frac{6.6}{\sqrt{14}} \approx 1.76. \][/tex]
Next, we calculate the z-score using the formula:
[tex]\[ z = \frac{\bar{x} - \mu_0}{\text{SEM}} = \frac{67.71 - 72}{1.76} \approx -2.43. \][/tex]
### Step 4: Determine the Critical Value
For a one-tailed test at the 0.05 significance level, the critical z-value is:
[tex]\[ z_{\alpha} = -1.645. \][/tex]
### Step 5: Make a Decision
To make a decision, we compare the calculated z-score with the critical z-value:
- If [tex]\( z \)[/tex] is less than [tex]\( z_{\alpha} \)[/tex], we reject [tex]\( H_0 \)[/tex].
- Observed z-score: -2.43
- Critical z-value: -1.645
Since [tex]\( -2.43 < -1.645 \)[/tex], we reject the null hypothesis.
### Conclusion
Given these results, we have sufficient evidence at the 0.05 significance level to support the claim that the sample pulse rates come from a population with a mean pulse rate less than 72 beats per minute.
### Step 1: State the Hypotheses
First, we need to set up our null and alternative hypotheses:
- [tex]\( H_0 \)[/tex]: The population mean pulse rate is equal to 72 beats per minute ([tex]\( \mu = 72 \)[/tex]).
- [tex]\( H_1 \)[/tex]: The population mean pulse rate is less than 72 beats per minute ([tex]\( \mu < 72 \)[/tex]).
Hence, the correct hypothesis from the provided choices is:
[tex]\[ H_0: \mu = 72 \text{ beats per minute} \][/tex]
[tex]\[ H_1: \mu < 72 \text{ beats per minute} \][/tex]
### Step 2: Gather Sample Data and Statistics
From the sample data provided, we have resting pulse rates:
[tex]\[ [56, 59, 69, 84, 74, 64, 69, 70, 66, 68, 59, 71, 76, 63]. \][/tex]
Given:
- Population standard deviation ([tex]\( \sigma \)[/tex]): 6.6 beats per minute
- Significance level ([tex]\( \alpha \)[/tex]): 0.05
For these data, we know:
- Sample size ([tex]\( n \)[/tex]): 14
- Sample mean ([tex]\( \bar{x} \)[/tex]): 67.71 beats per minute (approx.)
### Step 3: Calculate the Test Statistic
We calculate the standard error of the mean (SEM) using the formula:
[tex]\[ \text{SEM} = \frac{\sigma}{\sqrt{n}} = \frac{6.6}{\sqrt{14}} \approx 1.76. \][/tex]
Next, we calculate the z-score using the formula:
[tex]\[ z = \frac{\bar{x} - \mu_0}{\text{SEM}} = \frac{67.71 - 72}{1.76} \approx -2.43. \][/tex]
### Step 4: Determine the Critical Value
For a one-tailed test at the 0.05 significance level, the critical z-value is:
[tex]\[ z_{\alpha} = -1.645. \][/tex]
### Step 5: Make a Decision
To make a decision, we compare the calculated z-score with the critical z-value:
- If [tex]\( z \)[/tex] is less than [tex]\( z_{\alpha} \)[/tex], we reject [tex]\( H_0 \)[/tex].
- Observed z-score: -2.43
- Critical z-value: -1.645
Since [tex]\( -2.43 < -1.645 \)[/tex], we reject the null hypothesis.
### Conclusion
Given these results, we have sufficient evidence at the 0.05 significance level to support the claim that the sample pulse rates come from a population with a mean pulse rate less than 72 beats per minute.
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