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# In a order to compare the means of two populations, independent random samples of 410 observations are selected from each population, with Sample 1 the results found in the table to the right. Complete parts a through e below. X1 = 5,319 S1= 143 a. Use a 95% confidence interval to estimate the difference between the population means $H - H2$ Interpret the contidence interval. The contidence interval IS $Round to one decimal place as needed.$ Sample 2 X2 = 5,285 S2 = 198 Aa. Use a 95% confidence interval to estimate the difference between the population means $A1 - M2$ Interpret the contidence interval. The contidence interval Is $Round to one decimal place as needed.$ b. Test the null hypothesis Ho versus alternative hypothesis Ha $H What is the test statistic? H2$ + Give the significance level of the test, and interpret the result. Use a = 0.05. Z=

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## Answer to a math question In a order to compare the means of two populations, independent random samples of 410 observations are selected from each population, with Sample 1 the results found in the table to the right. Complete parts a through e below. X1 = 5,319 S1= 143 a. Use a 95% confidence interval to estimate the difference between the population means $H - H2$ Interpret the contidence interval. The contidence interval IS $Round to one decimal place as needed.$ Sample 2 X2 = 5,285 S2 = 198 Aa. Use a 95% confidence interval to estimate the difference between the population means $A1 - M2$ Interpret the contidence interval. The contidence interval Is $Round to one decimal place as needed.$ b. Test the null hypothesis Ho versus alternative hypothesis Ha $H What is the test statistic? H2$ + Give the significance level of the test, and interpret the result. Use a = 0.05. Z=

Darrell
4.5
To estimate the difference between the population means $μ1 - μ2$ using a 95% confidence interval, we can use the formula:

\text{{Confidence Interval}} = $\bar{x_1} - \bar{x_2}$ \pm Z \cdot \sqrt{\frac{{s_1^2}}{{n_1}} + \frac{{s_2^2}}{{n_2}}}

where:
- μ1 and μ2 are the population means
- \bar{x_1} and \bar{x_2} are the sample means
- s_1 and s_2 are the sample standard deviations
- n1 and n2 are the sample sizes
- Z is the Z-score for a 95% confidence interval, which is approximately 1.96

a. Using the given values:
\bar{x_1} = 5,319, s_1 = 143, n_1 = 410
\bar{x_2} = 5,285, s_2 = 198, n_2 = 410
Z = 1.96

Substituting these values into the formula:

\text{{Confidence Interval}} = $5,319 - 5,285$ \pm 1.96 \cdot \sqrt{\frac{{143^2}}{{410}} + \frac{{198^2}}{{410}}}

Simplifying the expression within the square root:

\sqrt{\frac{{143^2}}{{410}} + \frac{{198^2}}{{410}}} = \sqrt{\frac{{20,449}}{{410}}} + \sqrt{\frac{{39,204}}{{410}}}

\sqrt{\frac{{20,449 + 39,204}}{{410}}} = \sqrt{\frac{{59,653}}{{410}}} = \sqrt{\frac{{146}}{1}} \approx 12.083

Therefore, the confidence interval for the difference between the population means $μ1 - μ2$ is:

$5,319 - 5,285$ \pm 1.96 \cdot 12.083

34 \pm 23.733

The confidence interval for the difference in population means is $10.3, 57.7$. This means that we can be 95% confident that the true difference between the population means lies between 10.3 and 57.7.

b. To test the null hypothesis $H0: μ1 = μ2$ against the alternative hypothesis $Ha: μ1 ≠ μ2$, we can use a two-sample t-test. Since the sample sizes are large $both are 410$, we can approximate the test statistic using a Z-test.

The test statistic for a two-sample Z-test is given by:

Z = \frac{{$\bar{x_1} - \bar{x_2}$ - $\mu_1 - \mu_2$}}{{\sqrt{\frac{{s_1^2}}{{n_1}} + \frac{{s_2^2}}{{n_2}}}}}

where:
- μ1 and μ2 are the population means
- \bar{x_1} and \bar{x_2} are the sample means
- s_1 and s_2 are the sample standard deviations
- n1 and n2 are the sample sizes

Using the given values:
\bar{x_1} = 5,319, s_1 = 143, n_1 = 410
\bar{x_2} = 5,285, s_2 = 198, n_2 = 410

Substituting these values into the formula:

Z = \frac{{$5,319 - 5,285$ - 0}}{{\sqrt{\frac{{143^2}}{{410}} + \frac{{198^2}}{{410}}}}}

Simplifying the expression within the square root:

\sqrt{\frac{{143^2}}{{410}} + \frac{{198^2}}{{410}}} = \sqrt{\frac{{20,449}}{{410}}} + \sqrt{\frac{{39,204}}{{410}}}

\sqrt{\frac{{20,449 + 39,204}}{{410}}} = \sqrt{\frac{{59,653}}{{410}}} = \sqrt{\frac{{146}}{1}} \approx 12.083

So the test statistic Z is:

Z = \frac{{34}}{{12.083}} \approx 2.810

The test statistic is Z = 2.810. The significance level of the test is α = 0.05. Since the test statistic Z is greater than the critical value Zα/2 $approximately ±1.96 for a 95% confidence level$, we reject the null hypothesis. This means there is sufficient evidence to suggest that the population means are not equal.
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