A data set includes 106 body temperatures of healthy adult humans having a mean of and a standard deviation
of . Construct a confidence interval estimate of the mean body temperature of all healthy humans. What
does the sample suggest about the use of as the mean body temperature?
Click here to view a t distribution table.
Click here to view page 1 of the standard normal distribution table.
Click here to view page 2 of the standard normal distribution table.
What is the confidence interval estimate of the population mean ?

(Round to three decimal places as needed.)

Ask by Hammond Valdez. in the United States

Dec 09,2024

Question

A data set includes 106 body temperatures of healthy adult humans having a mean of and a standard deviation
of . Construct a confidence interval estimate of the mean body temperature of all healthy humans. What
does the sample suggest about the use of as the mean body temperature?
Click here to view a t distribution table.
Click here to view page 1 of the standard normal distribution table.
Click here to view page 2 of the standard normal distribution table.
What is the confidence interval estimate of the population mean ?

(Round to three decimal places as needed.)

Ask by Hammond Valdez. in the United States

Dec 09,2024

UpStudy AI · Accepted Answer

To construct a $$ 99\% $$ confidence interval for the mean body temperature of all healthy humans, we can use the formula for the confidence interval:

$$
\text{Confidence Interval} = \bar{x} \pm z^* \left( \frac{s}{\sqrt{n}} \right)
$$

where:
- $$\bar{x}$$ is the sample mean,
- $$z^*$$ is the z-score corresponding to the desired confidence level,
- $$s$$ is the sample standard deviation,
- $$n$$ is the sample size.

### Step 1: Identify the values
- Sample mean ($$\bar{x}$$) = $$98.9^{\circ} \mathrm{F}$$
- Sample standard deviation ($$s$$) = $$0.63^{\circ} \mathrm{F}$$
- Sample size ($$n$$) = $$106$$

### Step 2: Find the z-score for a $$99\%$$ confidence level
For a $$99\%$$ confidence level, the z-score ($$z^*$$) can be found using a z-table. The critical value for $$99\%$$ confidence is approximately $$2.576$$.

### Step 3: Calculate the standard error (SE)
$$
SE = \frac{s}{\sqrt{n}} = \frac{0.63}{\sqrt{106}} \approx \frac{0.63}{10.2956} \approx 0.0612
$$

### Step 4: Calculate the margin of error (ME)
$$
ME = z^* \cdot SE = 2.576 \cdot 0.0612 \approx 0.1575
$$

### Step 5: Construct the confidence interval
$$
\text{Lower limit} = \bar{x} - ME = 98.9 - 0.1575 \approx 98.7425
$$
$$
\text{Upper limit} = \bar{x} + ME = 98.9 + 0.1575 \approx 99.0575
$$

### Step 6: Round to three decimal places
$$
\text{Lower limit} \approx 98.743^{\circ} \mathrm{F}
$$
$$
\text{Upper limit} \approx 99.058^{\circ} \mathrm{F}
$$

### Final Confidence Interval
Thus, the $$99\%$$ confidence interval estimate of the population mean $$\mu$$ is:

$$
98.743^{\circ} \mathrm{F} < \mu < 99.058^{\circ} \mathrm{F}
$$

### Conclusion about $$98.6^{\circ} \mathrm{F}$$
Since $$98.6^{\circ} \mathrm{F}$$ falls within the confidence interval $$ (98.743^{\circ} \mathrm{F}, 99.058^{\circ} \mathrm{F}) $$, the sample suggests that $$98.6^{\circ} \mathrm{F}$$ could still be a reasonable estimate for the mean body temperature of healthy adults, but it is slightly lower than the lower limit of the confidence interval.
The $$99\%$$ confidence interval estimate of the population mean $$\mu$$ is $$98.743^{\circ} \mathrm{F} < \mu < 99.058^{\circ} \mathrm{F}$$. The sample suggests that $$98.6^{\circ} \mathrm{F}$$ could be a reasonable estimate for the mean body temperature, but it is slightly lower than the lower limit of the confidence interval.

Upstudy AI Solution

Answer

Solution

Step 1: Identify the values

Step 2: Find the z-score for a confidence level

Step 3: Calculate the standard error (SE)

Step 4: Calculate the margin of error (ME)

Step 5: Construct the confidence interval

Step 6: Round to three decimal places

Final Confidence Interval

Conclusion about

Extra Insights

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