Answer:
4.08 < μ1 − μ2 < 5.94
Step-by-step explanation:
The confidence level interval for μ is given by;
= ⁻x ± z*[tex]\frac{σ}{\sqrt{n} }[/tex]
where n = Sample size
σ = Standard deviation
⁻x = Mean
For a Confidence level = 94%, the critical value is z = 1.88
Considering the first random sample:
n1 = 25
σ1 = 5
⁻x1 = 80
The confidence level interval for μ1 is given by;
μ1 = [tex]80[/tex] ± 1.88 × [tex]\frac{5}{ \sqrt{25} }[/tex]
μ1 = 80 ± (1.88 ×1) = 80 ± 1.88
μ1 = 78.12 < μ1 < 81.88
For the second random sample
n2 = 36
σ2 = 3
⁻x2 = 75
The confidence level interval for μ2 is given by;
μ2 = [tex]75[/tex] ± 1.88 × [tex]\frac{3}{ \sqrt{36} }[/tex]
μ2 = 75 ± (1.88 × 0.5) = 75 ± 0.94
μ2 = 74.04 < μ2 < 75.94
For the interval μ1 − μ2,
The confidence level is 78.12 - 74.04 < μ1 − μ2 < 81.88 - 75.94
4.08 < μ1 − μ2 < 5.94
A 94% confidence interval for μ1 − μ2 of the random sample is [tex]4.06 < \mu_1 - \mu_2 < 5.94[/tex]
The given parameters are:
Sample size 1, [tex]n_1 = 25[/tex]
Standard deviation 1, [tex]\sigma_1 = 5\\[/tex]
Mean 1, [tex]\bar x_1 = 80[/tex]
Sample size 2, [tex]n_2 = 36[/tex]
Standard deviation 2, [tex]\sigma_2= 3[/tex]
Mean 2, [tex]\bar x_2 = 75[/tex]
The confidence interval for the mean is calculated by:
[tex]CI = \bar x \pm z* \frac{\sigma}{\sqrt n}[/tex]
At 94% confidence interval, the critical value is:
[tex]z = 1.88[/tex]
For the first random sample, we have:
[tex]CI_1 = 80 \pm 1.88 * \frac{5}{\sqrt {25}}[/tex]
[tex]CI_1 = 80 \pm 1.88[/tex]
Expand
[tex]CI_1 = (80 - 1.88, 80 + 1.88)[/tex]
[tex]CI_1 = (78.12, 81.88)[/tex]
For the second random sample, we have:
[tex]CI_2 = 75 \pm 1.88 * \frac{3}{\sqrt {36}}[/tex]
[tex]CI_2 = 75 \pm 1.88 * 0.5[/tex]
[tex]CI_2 = 75 \pm 0.94[/tex]
Expand
[tex]CI_2 = (75 - 0.94, 75 + 0.94)[/tex]
[tex]CI_2 = (74.06, 75.94)[/tex]
Calculate the difference
[tex]CI_1 - CI_2 = (78.12 - 74.06, 81.88 - 75.94)[/tex]
Evaluate the difference
[tex]CI_1 - CI_2 = (4.06, 5.94)[/tex]
Rewrite as:
[tex]4.06 < \mu_1 - \mu_2 < 5.94[/tex]
Hence, a 94% confidence interval for μ1 − μ2 is [tex]4.06 < \mu_1 - \mu_2 < 5.94[/tex]
Read more about confidence intervals at:
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