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Exam 7: Estimating Parameters and Determining Sample Sizes

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Question 31

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Define margin of error. Explain the relation between the confidence interval and the margin of error. Suppose a confidence interval is $9.65 < \mu < 11.35$ Redefine the confidence interval into a format using the margin of error and the point estimate of the population mean.

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The margin of error is the maximum likel...

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Question 32

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Use the given degree of confidence and sample data to construct a confidence interval for the population proportion p. n=130, x=69 ; 90% confidence

A) $0.458 < p < 0.604$
B) $0.461 < p < 0.601$
C) $0.459 < p < 0.603$
D) $0.463 < p < 0.599$

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Question 33

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Describe the steps for finding a confidence interval

In general, what does "degrees of freedom" refer to? Find the degrees of freedom for the given information, assuming that you want to construct a confidence interval estimate of $\mu :$ Six human skulls from around 4000 B.C. were measured, and the lengths have a mean of 94.2 mm and a standard deviation of 4.9 mm.

Find the degree of confidence used in constructing the confidence interval $0.523 < p < 0.669$ - for the population proportion p using sample data with n=109, x=65 .

Interpret the following 95 % confidence interval for mean weekly salaries of shift managers at Guiseppe's Pizza and Pasta. $325.80 < \mu < 472.30$

Which of the following critical values is appropriate for a 98% confidence level where n=7 ; $\sigma = 27$ and the population appears to be normally distributed.

Find the critical value $z _ { \alpha / 2 }$ that corresponds to a 91% confidence level.

Find the value of $z _ { \alpha / 2 }$ that corresponds to a confidence level of 89.48%.

Use the given degree of confidence and sample data to construct a confidence interval for the population mean $\mu .$ Assume that the population has a normal distribution. A laboratory tested twelve chicken eggs and found that the mean amount of cholesterol was 185 milligrams with s=17.6 milligrams. A confidence interval of $173.8 \mathrm { mg } < \mu < 196.2$ mg is constructed for the true mean cholesterol content of all such eggs. It was assumed that the population has a normal distribution. What confidence level does this interval represent?

Express a confidence interval defined as (0.432,0.52) in the form of the point estimate _____ $\pm$ the margin of error______ Express both in three decimal places.

Identify the correct distribution (z, t, or neither)for each of the following $\begin{array}{l}\text { Identify the correct distribution } ( z , t \text {, or neither) for each of the following. }\\\begin{array} { | l | l | l | l | } \hline \text { Sample Size } & \text { Standard Deviation } & \begin{array} { l } \text { Shape of the } \\\text { Distribution }\end{array} & z \text { or } t \text { or neither } \\\hline n = 35 & s = 4.5 & \text { Somewhat skewed } & \\\hline n = 20 & s = 4.5 & \text { Bell shaped } & \\\hline n = 35 & \sigma = 4.5 & \text { Bell shaped } & \\\hline n = 20 & \sigma = 4.5 & \text { Extremely skewed } & \\\hline\end{array}\end{array}$

Draw a diagram of the chi-square distribution. Discuss its shape and values

In constructing a confidence interval for $\sigma$ or $\sigma ^ { 2 }$ , a table is used to find the critical values $\chi _ { \mathrm { L } } ^ { 2 } \text { and } \chi _ { \mathrm { R } } ^ { 2 }$ for values of $n \leq 101$ For larger values of n, $\chi _ { \mathrm { L } } ^ { 2 } \text { and } \chi _ { \mathrm { R } } ^ { 2 }$ can be approximated by using the following formula: $\chi ^ { 2 } = \frac { 1 } { 2 } \pm z _ { \alpha / 2 } + \sqrt { 2 k - 1 } ^ { 2 }$ where k is the number of degrees of freedom and $z _ { \alpha / 2 }$ is the critical z-score. Construct the 90 % confidence interval for $\sigma$ using the following sample data: a sample of size n=232 yields a mean weight of 154 lb and a standard deviation of 25.5 lb. Round the confidence interval limits to the nearest hundredth.

Use the given degree of confidence and sample data to construct a confidence interval for the population mean $\mu$ Assume that the population has a normal distribution. A laboratory tested twelve chicken eggs and found that the mean amount of cholesterol was 185 milligrams with s=17.6 milligrams. Construct a 95% confidence interval for the true mean cholesterol content of all such eggs.

Use the given degree of confidence and sample data to construct a confidence interval for the population mean μ. Assume that the population has a normal distribution. Thirty randomly selected students took the calculus final. If the sample mean was 95 and the standard deviation was $6.6$ , construct a $99 \%$ confidence interval for the mean score of all students.

The margin of error ________ ________ (increases or decreases) with an increase in confidence level.

Assume that a sample is used to estimate a population proportion p. Find the margin of error E that corresponds to the given statistics and confidence level. Round the margin of error to four decimal places. 98% confidence; the sample size is 800, of which 40% are successes

To find the standard deviation of the diameter of wooden dowels, the manufacturer measures 19 randomly selected dowels and finds the standard deviation of the sample to be s=0.16 . Find the 95 % confidence interval for the population standard deviation $\sigma$

Express the confidence interval $0.039 < p < 0.479$ in the form of $\hat { p } \pm \mathrm { E }$