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Answer:
the 90% confidence interval for the mean number of years people use a two-wheeler after purchasing it is [tex]\( (11.213 to 13.187) \)[/tex]years
Step-by-step explanation:
To find the 90% confidence interval for the mean number of years people use a two-wheeler after purchasing it, we can use the formula for the confidence interval:
[tex]\[ \text{Confidence Interval} = \bar{x} \pm Z \cdot \frac{s}{\sqrt{n}} \][/tex]
where:
-[tex]\( \bar{x} \)[/tex] is the sample mean,
- [tex]\( s \)[/tex] is the sample standard deviation,
- [tex]\( n \)[/tex] is the sample size,
- [tex]\( Z \)[/tex] is the critical value from the standard normal distribution corresponding to the desired confidence level.
Given:
- Sample mean [tex]\( \bar{x} = 12.20 \)[/tex] years,
- Sample standard deviation[tex]\( s = 6 \)[/tex]years,
- Sample size[tex]\( n = 100 \)[/tex],
- Desired confidence level [tex]\( 90\% \)[/tex].
1. Find the critical value [tex]\( Z \):[/tex]
- For a 90% confidence level, the critical value [tex]\( Z \) is \( 1.645 \)[/tex].
2. Calculate the margin of error [tex]\( E \):[/tex]
[tex]\[ E = Z \cdot \frac{s}{\sqrt{n}} \] \[ E = 1.645 \cdot \frac{6}{\sqrt{100}} \] \[ E = 1.645 \cdot \frac{6}{10} \] \[ E = 1.645 \cdot 0.6 \] \[ E = 0.987 \][/tex]
3. Construct the confidence interval:
[tex]\[ \text{Confidence Interval} = \bar{x} \pm E \][/tex]
[tex]\[ \text{Confidence Interval} = 12.20 \pm 0.987 \][/tex]
[tex]\[ \text{Confidence Interval} = (11.213, 13.187) \][/tex]
Therefore, the 90% confidence interval for the mean number of years people use a two-wheeler after purchasing it is [tex]\( (11.213, 13.187) \)[/tex] years. This means we are 90% confident that the true population mean of years people use a two-wheeler after purchase lies within this interval.
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