Answer :
To construct an 80% confidence interval for the population mean, we start with the following information:
- Sample mean: [tex]$$\bar{x} = 1.69$$[/tex]
- Population standard deviation: [tex]$$\sigma = 0.657$$[/tex]
- Sample size: [tex]$$n = 50$$[/tex]
Step 1. Determine the critical [tex]$z$[/tex]-value.
For an 80% confidence interval, the total confidence in both tails is [tex]$$1 - 0.80 = 0.20.$$[/tex] Since the tail areas are symmetric, each tail has an area of [tex]$$\frac{0.20}{2} = 0.10.$$[/tex] Therefore, the critical [tex]$z$[/tex]-value corresponds to the 90th percentile of the standard normal distribution:
[tex]$$ z_{0.90} \approx 1.2816. $$[/tex]
Step 2. Compute the standard error (SE) of the sample mean.
The standard error is given by:
[tex]$$ SE = \frac{\sigma}{\sqrt{n}}. $$[/tex]
Substituting the values:
[tex]$$ SE = \frac{0.657}{\sqrt{50}} \approx 0.09291. $$[/tex]
Step 3. Calculate the margin of error (ME).
The margin of error is given by:
[tex]$$ ME = z \times SE. $$[/tex]
Substituting the values:
[tex]$$ ME = 1.2816 \times 0.09291 \approx 0.11907. $$[/tex]
Step 4. Construct the confidence interval.
The confidence interval for the population mean is given by:
[tex]$$ \bar{x} \pm ME. $$[/tex]
Thus, the interval is:
[tex]$$ 1.69 \pm 0.11907. $$[/tex]
Rounded to the precision given in the options, this is:
[tex]$$ 1.69 \pm 0.119. $$[/tex]
Final Answer:
The 80% confidence interval for the population mean is:
[tex]$$ 1.69 \pm 0.119. $$[/tex]
- Sample mean: [tex]$$\bar{x} = 1.69$$[/tex]
- Population standard deviation: [tex]$$\sigma = 0.657$$[/tex]
- Sample size: [tex]$$n = 50$$[/tex]
Step 1. Determine the critical [tex]$z$[/tex]-value.
For an 80% confidence interval, the total confidence in both tails is [tex]$$1 - 0.80 = 0.20.$$[/tex] Since the tail areas are symmetric, each tail has an area of [tex]$$\frac{0.20}{2} = 0.10.$$[/tex] Therefore, the critical [tex]$z$[/tex]-value corresponds to the 90th percentile of the standard normal distribution:
[tex]$$ z_{0.90} \approx 1.2816. $$[/tex]
Step 2. Compute the standard error (SE) of the sample mean.
The standard error is given by:
[tex]$$ SE = \frac{\sigma}{\sqrt{n}}. $$[/tex]
Substituting the values:
[tex]$$ SE = \frac{0.657}{\sqrt{50}} \approx 0.09291. $$[/tex]
Step 3. Calculate the margin of error (ME).
The margin of error is given by:
[tex]$$ ME = z \times SE. $$[/tex]
Substituting the values:
[tex]$$ ME = 1.2816 \times 0.09291 \approx 0.11907. $$[/tex]
Step 4. Construct the confidence interval.
The confidence interval for the population mean is given by:
[tex]$$ \bar{x} \pm ME. $$[/tex]
Thus, the interval is:
[tex]$$ 1.69 \pm 0.11907. $$[/tex]
Rounded to the precision given in the options, this is:
[tex]$$ 1.69 \pm 0.119. $$[/tex]
Final Answer:
The 80% confidence interval for the population mean is:
[tex]$$ 1.69 \pm 0.119. $$[/tex]
