Questions: If n=10, x̄(x-bar)=36, and s=7, construct a confidence interval at a 80% confidence level. Assume the data came from a normally distributed population. Give your answers to one decimal place. <μ<

Solution Steps

To construct a confidence interval for the population mean when the sample size is small and the population standard deviation is unknown, we use the t-distribution. Given the sample size $n = 10$, sample mean $\bar{x}$, sample standard deviation $s = 7$, and the confidence level of 80%, we can find the t-critical value from the t-distribution table. The confidence interval is then calculated using the formula:

$$\bar{x} \pm t^* \left(\frac{s}{\sqrt{n}}\right)$$

where $t^*$ is the t-critical value for 9 degrees of freedom (since $n-1 = 9$) at the 80% confidence level.

We are given the following values:

• Sample size: $n = 10$
• Sample standard deviation: $s = 7$
• Confidence level: $80\%$
• The expression $\bar{x}(\mathrm{x}-\mathrm{bar}) = 36$ implies that the sample mean is calculated as $\bar{x} = \frac{36}{n} = \frac{36}{10} = 3.6$.

The degrees of freedom for our sample is calculated as: $$df = n - 1 = 10 - 1 = 9$$ Using the t-distribution for $df = 9$ at a confidence level of $80\%$, we find the t-critical value: $$t^* \approx 1.3830$$

The margin of error (ME) is calculated using the formula: $$ME = t^* \left(\frac{s}{\sqrt{n}}\right)$$ Substituting the values: $$ME \approx 1.3830 \left(\frac{7}{\sqrt{10}}\right) \approx 3.0615$$

The confidence interval for the population mean $\mu$ is given by: $$\bar{x} \pm ME$$ Calculating the lower and upper bounds: $$\text{Lower Bound} = \bar{x} - ME \approx 3.6 - 3.0615 \approx 0.5385$$ $$\text{Upper Bound} = \bar{x} + ME \approx 3.6 + 3.0615 \approx 6.6615$$

Final Answer