Formula tamaño muestra:


N=	180
MU=	20158
SIGMA=	7380
e%	5%
e=	9
NC=	95%
z_a/2=	0,06
Millaje
 	L inf	L sup	# datos	%
1	12778	27558	119	66%
2	5399	34958	172	96%
3	-1981	42359	178	99%
4. Construyan un intervalo de confianza para determinar si el millaje recorrido por los autos fabricados por Cadillac y Chevrolet es el mismo o es diferente. Ahora, asuman que deben privilegiar la confianza sobre la exactitud. Justifiquen su elección del nivel de confianza. Asuma que las medias y las varianzas poblacionales son desconocidas.

Registramos los datos obtenidos en Excel para la marca Chevrolet

mu	19656
n	80
xBar	17874
s	8600
nc	99%
gl	79
a/2	0,005
t_a/2	2,64
errEst	962
e	2538
Linf	15336
Lsup	20412
Tome el 99% de confianza para que la media entrara en el rango de limite superior e inferior.

Registramos los datos obtenidos en Excel para la marca Cadillac

mu	18909
n	20
xBar	19199
s	7303
nc	95%
gl	19
a/2	0,025
t_a/2	2,09
errEst	1633
e	3418
Linf	15781
Lsup	22617

### Construction of Confidence Intervals

We are tasked with determining whether the mileage traveled by cars manufactured by Cadillac and Chevrolet is the same or different, using a confidence interval (CI) analysis. The idea is to compare the confidence intervals of both brands based on sample data and infer whether their true population means differ significantly. Let's break this down:

### Chevrolet Confidence Interval
- Sample mean \(\bar{x}_{Chevy} = 17874\)
- Sample size \(n_{Chevy} = 80\)
- Sample standard deviation \(s_{Chevy} = 8600\)
- Confidence level \(nc = 99\%\)
- Degrees of freedom \(df = 79\)
- Critical value \(t_{\alpha/2} = 2.64\) (from the t-distribution table)
- Standard error of the mean:
\[
SE = \frac{s_{Chevy}}{\sqrt{n}} = \frac{8600}{\sqrt{80}} = 962
\]
- Margin of error \(E = t_{\alpha/2} \times SE = 2.64 \times 962 = 2538\)

So, the 99% confidence interval for Chevrolet is:
\[
CI_{Chevy} = \left( \bar{x}_{Chevy} - E, \bar{x}_{Chevy} + E \right) = \left(17874 - 2538, 17874 + 2538\right) = (15336, 20412)
\]

### Cadillac Confidence Interval
- Sample mean \(\bar{x}_{Cadillac} = 19199\)
- Sample size \(n_{Cadillac} = 20\)
- Sample standard deviation \(s_{Cadillac} = 7303\)
- Confidence level \(nc = 95\%\)
- Degrees of freedom \(df = 19\)
- Critical value \(t_{\alpha/2} = 2.09\) (from the t-distribution table)
- Standard error of the mean:
\[
SE = \frac{s_{Cadillac}}{\sqrt{n}} = \frac{7303}{\sqrt{20}} = 1633
\]
- Margin of error \(E = t_{\alpha/2} \times SE = 2.09 \times 1633 = 3418\)

So, the 95% confidence interval for Cadillac is:
\[
CI_{Cadillac} = \left( \bar{x}_{Cadillac} - E, \bar{x}_{Cadillac} + E \right) = \left(19199 - 3418, 19199 + 3418\right) = (15781, 22617)
\]

### Conclusion
The confidence intervals for Chevrolet \((15336, 20412)\) and Cadillac \((15781, 22617)\) overlap slightly, meaning that it's plausible that the true mean mileage for the two car brands could be the same. However, since the intervals are quite close to not overlapping, it suggests that further testing, possibly with larger sample sizes or a different confidence level, might reveal more significant differences.

### Justification for Confidence Level
You chose a 99% confidence level for Chevrolet and a 95% confidence level for Cadillac. The 99% confidence level is justified if you want to prioritize confidence in your estimates over precision (a wider interval but higher certainty). In contrast, a 95% confidence level sacrifices some confidence for a narrower interval, which provides more precise but slightly less certain estimates.

If your goal is to privilege confidence over exactness, as stated, using the 99% confidence level is appropriate. You might consider using it consistently across both comparisons to maintain a balance in the analysis.

---

Would you like more details on any specific part? Here are some related questions:

1. How can the sample size influence the confidence intervals?
2. What is the significance of overlapping confidence intervals?
3. How does changing the confidence level affect the width of the interval?
4. What other statistical tests can be used to compare the two brands' mileages?
5. Why is the t-distribution used here instead of the z-distribution?

**Tip**: When comparing two groups, overlapping confidence intervals suggest there may be no significant difference, but it’s always good to verify with additional tests like a two-sample t-test.

Formula tamaño muestra:

N= 180
MU= 20158
SIGMA= 7380
e% 5%
e= 9
NC= 95%
z_a/2= 0,06
Millaje
 	L inf	L sup	# datos	%
1	12778	27558	119	66%
2	5399	34958	172	96%
3	-1981	42359	178	99%
4. Construyan un intervalo de confianza para determinar si el millaje recorrido por los autos fabricados por Cadillac y Chevrolet es el mismo o es diferente. Ahora, asuman que deben privilegiar la confianza sobre la exactitud. Justifiquen su elección del nivel de confianza. Asuma que las medias y las varianzas poblacionales son desconocidas.

Registramos los datos obtenidos en Excel para la marca Chevrolet

mu 19656
n 80
xBar 17874
s 8600
nc 99%
gl 79
a/2 0,005
t_a/2 2,64
errEst 962
e 2538
Linf 15336
Lsup 20412
Tome el 99% de confianza para que la media entrara en el rango de limite superior e inferior.

Registramos los datos obtenidos en Excel para la marca Cadillac

mu 18909
n 20
xBar 19199
s 7303
nc 95%
gl 19
a/2 0,025
t_a/2 2,09
errEst 1633
e 3418
Linf 15781
Lsup 22617

Analyze whether the mileage of Cadillac and Chevrolet cars is statistically different using confidence intervals. Learn how to construct confidence intervals for both brands with a focus on 99% and 95% confidence levels. Suitable for university-level statistics students.

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