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Sagot :
To construct a 95% confidence interval for [tex]\( \mu_d \)[/tex], the population mean difference in tread wear between the two brands, we will follow these steps:
1. Calculate the mean of the sample differences:
[tex]\[ \bar{d} = \text{mean of the differences} \][/tex]
Given the sample differences are:
[tex]\[ [-0.3, 0.34, -0.9, 0.03, -1.03, 0.12, -0.12, -0.15, -0.28, -0.02, -0.25, -0.74, -0.32] \][/tex]
We obtain:
[tex]\[ \bar{d} = -0.278 \][/tex]
2. Calculate the standard deviation of the sample differences:
[tex]\[ s_d = \text{standard deviation of the differences} \][/tex]
We obtain:
[tex]\[ s_d = 0.399 \][/tex]
3. Calculate the standard error of the mean difference:
[tex]\[ SE = \frac{s_d}{\sqrt{n}} \][/tex]
where [tex]\( n \)[/tex] is the number of differences. For our sample:
[tex]\[ n = 13 \][/tex]
We obtain:
[tex]\[ SE = \frac{0.399}{\sqrt{13}} \approx 0.111 \][/tex]
4. Determine the critical value:
For a 95% confidence interval and [tex]\( n - 1 = 12 \)[/tex] degrees of freedom, the critical value ([tex]\( t \)[/tex] value) from the t-distribution is:
[tex]\[ t_{\alpha/2} \approx 2.179 \][/tex]
5. Calculate the margin of error:
[tex]\[ ME = t_{\alpha/2} \times SE \][/tex]
We obtain:
[tex]\[ ME = 2.179 \times 0.111 \approx 0.241 \][/tex]
6. Calculate the confidence interval:
[tex]\[ \text{Lower limit} = \bar{d} - ME \][/tex]
[tex]\[ \text{Upper limit} = \bar{d} + ME \][/tex]
We obtain:
[tex]\[ \text{Lower limit} = -0.278 - 0.241 \approx -0.520 \][/tex]
[tex]\[ \text{Upper limit} = -0.278 + 0.241 \approx -0.037 \][/tex]
Therefore, the 95% confidence interval for the population mean difference in tread wear between the two brands is approximately:
[tex]\[ \text{Lower limit}: -0.52 \][/tex]
[tex]\[ \text{Upper limit}: -0.04 \][/tex]
1. Calculate the mean of the sample differences:
[tex]\[ \bar{d} = \text{mean of the differences} \][/tex]
Given the sample differences are:
[tex]\[ [-0.3, 0.34, -0.9, 0.03, -1.03, 0.12, -0.12, -0.15, -0.28, -0.02, -0.25, -0.74, -0.32] \][/tex]
We obtain:
[tex]\[ \bar{d} = -0.278 \][/tex]
2. Calculate the standard deviation of the sample differences:
[tex]\[ s_d = \text{standard deviation of the differences} \][/tex]
We obtain:
[tex]\[ s_d = 0.399 \][/tex]
3. Calculate the standard error of the mean difference:
[tex]\[ SE = \frac{s_d}{\sqrt{n}} \][/tex]
where [tex]\( n \)[/tex] is the number of differences. For our sample:
[tex]\[ n = 13 \][/tex]
We obtain:
[tex]\[ SE = \frac{0.399}{\sqrt{13}} \approx 0.111 \][/tex]
4. Determine the critical value:
For a 95% confidence interval and [tex]\( n - 1 = 12 \)[/tex] degrees of freedom, the critical value ([tex]\( t \)[/tex] value) from the t-distribution is:
[tex]\[ t_{\alpha/2} \approx 2.179 \][/tex]
5. Calculate the margin of error:
[tex]\[ ME = t_{\alpha/2} \times SE \][/tex]
We obtain:
[tex]\[ ME = 2.179 \times 0.111 \approx 0.241 \][/tex]
6. Calculate the confidence interval:
[tex]\[ \text{Lower limit} = \bar{d} - ME \][/tex]
[tex]\[ \text{Upper limit} = \bar{d} + ME \][/tex]
We obtain:
[tex]\[ \text{Lower limit} = -0.278 - 0.241 \approx -0.520 \][/tex]
[tex]\[ \text{Upper limit} = -0.278 + 0.241 \approx -0.037 \][/tex]
Therefore, the 95% confidence interval for the population mean difference in tread wear between the two brands is approximately:
[tex]\[ \text{Lower limit}: -0.52 \][/tex]
[tex]\[ \text{Upper limit}: -0.04 \][/tex]
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