# Solve and answer this question Are the results between the two confidence intervals very different 27.2 hg < mu < 29.4 hg What is the confidence interval for the population mean mu?

Question
Confidence intervals
Solve and answer this question Are the results between the two confidence intervals very different
$$27.2 hg < \mu < 29.4 hg$$
What is the confidence interval for the population mean $$\mu?$$

2020-11-09
From the provided information,
Sample size $$(n) = 244$$
Sample mean $$(\overline{x}) = 28.3 hg$$
Sample standard deviation $$(s) = 7.1 hg$$
Since, the population standard deviation is unknown, therefore, the t distribution will be used.
Confidence level $$= 98\%$$
Level of significance $$(\alpha) = 1 – 0.98 = 0.02$$
The degree of freedom = n – 1 = 244 – 1 = 243
The critical value of t at 243 degree of freedom with 0.01 level of significance from the t value table is 2.34.
The required $$98\%$$ confidence interval can be obtained as:
$$CI = \overline{x} \pm t_{\alpha/2,n-1}\frac{S}{\sqrt{n}}$$
$$= 28.3 \pm (2.34) \frac{7.1}{\sqrt{244}}$$
$$= 28.3 \pm 1.1$$
$$= (27. 2, 29.4)$$
Thus, the confidence interval is $$27.2 < \mu < 29.4.$$
No, the results between the two confidence intervals are not very different.
The confidence interval limits are almost similar. Therefore, the correct option is B).

### Relevant Questions

Here are summary stastistics for randomly selected weights of newborn girls: $$\displaystyle{n}={224},\overline{{{x}}}={28.3}\text{hg},{s}={7.1}\text{hg}$$. Construct a confidence interval estimate of mean. Use a 98% confidence level. Are these results very different from the confidence interval $$\displaystyle{26.5}\text{hg}{<}\mu{<}{30.7}\text{hg}$$ with only 14 sample values, $$\displaystyle\overline{{{x}}}={28.6}$$ hg, and $$\displaystyle{s}={2.9}$$ hg? What is the confidence interval for the population mean $$\displaystyle\mu$$? $$\displaystyle?{<}\mu{<}?$$ Are the results between the two confidence intervals very different?

A random sample of $$n_1 = 14$$ winter days in Denver gave a sample mean pollution index $$x_1 = 43$$.
Previous studies show that $$\sigma_1 = 19$$.
For Englewood (a suburb of Denver), a random sample of $$n_2 = 12$$ winter days gave a sample mean pollution index of $$x_2 = 37$$.
Previous studies show that $$\sigma_2 = 13$$.
Assume the pollution index is normally distributed in both Englewood and Denver.
(a) State the null and alternate hypotheses.
$$H_0:\mu_1=\mu_2.\mu_1>\mu_2$$
$$H_0:\mu_1<\mu_2.\mu_1=\mu_2$$
$$H_0:\mu_1=\mu_2.\mu_1<\mu_2$$
$$H_0:\mu_1=\mu_2.\mu_1\neq\mu_2$$
(b) What sampling distribution will you use? What assumptions are you making? NKS The Student's t. We assume that both population distributions are approximately normal with known standard deviations.
The standard normal. We assume that both population distributions are approximately normal with unknown standard deviations.
The standard normal. We assume that both population distributions are approximately normal with known standard deviations.
The Student's t. We assume that both population distributions are approximately normal with unknown standard deviations.
(c) What is the value of the sample test statistic? Compute the corresponding z or t value as appropriate.
(Test the difference $$\mu_1 - \mu_2$$. Round your answer to two decimal places.) NKS (d) Find (or estimate) the P-value. (Round your answer to four decimal places.)
(e) Based on your answers in parts (i)−(iii), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level \alpha?
At the $$\alpha = 0.01$$ level, we fail to reject the null hypothesis and conclude the data are not statistically significant.
At the $$\alpha = 0.01$$ level, we reject the null hypothesis and conclude the data are statistically significant.
At the $$\alpha = 0.01$$ level, we fail to reject the null hypothesis and conclude the data are statistically significant.
At the $$\alpha = 0.01$$ level, we reject the null hypothesis and conclude the data are not statistically significant.
(f) Interpret your conclusion in the context of the application.
Reject the null hypothesis, there is insufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Reject the null hypothesis, there is sufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Fail to reject the null hypothesis, there is insufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Fail to reject the null hypothesis, there is sufficient evidence that there is a difference in mean pollution index for Englewood and Denver. (g) Find a 99% confidence interval for
$$\mu_1 - \mu_2$$.
lower limit
upper limit
(h) Explain the meaning of the confidence interval in the context of the problem.
Because the interval contains only positive numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is greater than that of Denver.
Because the interval contains both positive and negative numbers, this indicates that at the 99% confidence level, we can not say that the mean population pollution index for Englewood is different than that of Denver.
Because the interval contains both positive and negative numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is greater than that of Denver.
Because the interval contains only negative numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is less than that of Denver.
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