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<

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.

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Transcript text: If $\mathrm{n}=10, \bar{x}(\mathrm{x}-\mathrm{bar})=36$, 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. $\square$ $<\mu<$ $\square$

Solution

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.

Step 1: Given Values

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 $.

Step 2: Degrees of Freedom and t-Critical Value

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: \[ t^* \approx 1.3830 \]

Step 3: Margin of Error Calculation

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

Step 4: Confidence Interval Calculation

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