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Problem I. One measure of a company's financial health is its debt-to- equity ratio. This quantity is defined to be the ratio of the company's corporate debt to the company's equity. If this ratio is to high, it is one indication of financial instability. Banks often monitor the financial health of companies to which they have extended commercial loans. A "lean" debt-to-equity ratio is considered to be a ratio of 1.50. A particular bank ran an audit for its loan portfolio's debt-to-equity ratio and randomly selected a sample of 15 of its commercial loan accounts, producing the follow result: 1.32 1.27 1.45 1.67 1.73 1.49 1.59 1.09 1.26 1.78 1.36 1.65 1.54 1.69 1.38 The sample mean and standard deviation for above sample are, A. What is the point estimator of population mean, ? B. Find a 98% confidence interval for the bank's debt-to-equity ratio for its loan portfolio. c. Interpret the results you found in A. D. How would the confidence interval change if you had assumed that the standard deviation was known to be 0.202? 

          Problem I. One measure of a company's financial health is its debt-to-
equity ratio. This quantity is defined to be the ratio of the company's
corporate debt to the company's equity. If this ratio is to high, it is one
indication of financial instability. Banks often monitor the financial health
of companies to which they have extended commercial loans. A "lean"
debt-to-equity ratio is considered to be a ratio of 1.50. A particular bank
ran an audit for its loan portfolio's debt-to-equity ratio and randomly
selected a sample of 15 of its commercial loan accounts, producing the
follow result:
1.32 1.27 1.45 1.67 1.73
1.49 1.59 1.09 1.26 1.78
1.36 1.65 1.54 1.69 1.38

The sample mean and standard deviation for above sample are,
A. What is the point estimator of population mean, ?

B. Find a 98% confidence interval for the bank's debt-to-equity ratio for its
loan portfolio.

c. Interpret the results you found in A.
D. How would the confidence interval change if you had assumed that the
standard deviation was known to be 0.202?
Show moreā€¦
Problem I. One measure of a company's financial health is its debt-to- equity ratio. This quantity is defined to be the ratio of the company's corporate debt to the company's equity. If this ratio is to high, it is one indication of financial instability. Banks often monitor the financial health of companies to which they have extended commercial loans. A "lean" debt-to-equity ratio is considered to be a ratio of 1.50. A particular bank ran an audit for its loan portfolio's debt-to-equity ratio and randomly selected a sample of 15 of its commercial loan accounts, producing the follow result: 1.32 1.27 1.45 1.67 1.73 1.49 1.59 1.09 1.26 1.78 1.36 1.65 1.54 1.69 1.38 The sample mean and standard deviation for above sample are, A. What is the point estimator of population mean, ? B. Find a 98% confidence interval for the bank's debt-to-equity ratio for its loan portfolio. c. Interpret the results you found in A. D. How would the confidence interval change if you had assumed that the standard deviation was known to be 0.202?

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Elementary Statistics a Step by Step Approach
Elementary Statistics a Step by Step Approach
Allan G. Bluman 9th Edition
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Problem I. One measure of a company's financial health is its debt-to-equity ratio. This quantity is defined to be the ratio of the company's corporate debt to the company's equity. If this ratio is to high, it is one indication of financial instability. Banks often monitor the financial health of companies to which they have extended commercial loans. A ā€leanā€ debt-to-equity ratio is considered to be a ratio of 1.50. A particular bank ran an audit for its loan portfolio's debt-to-equity ratio and randomly selected a sample of 15 of its commercial loan accounts, producing the follow result: 1.32 1.27 1.45 1.67 1.73 1.49 1.59 1.09 1.26 1.78 1.36 1.65 1.54 1.69 1.38. The sample mean and standard deviation for above sample are, A. What is the point estimator of population mean, ? B. Find a 98% confidence interval for the bank's debt-to-equity ratio for its loan portfolio. c. Interpret the results you found in A. D. How would the confidence interval change if you had assumed that the standard deviation was known to be 0.202?
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Transcript

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00:01 So in item a, we should give an estimate of the estimator for the mean here.
00:09 So basically, the estimate for the population mean is the simple mean.
00:15 So what we need to do is sum this observation, this 15 observations that we have, and divide this sum by 15.
00:22 And we are going to get that the mean is 148, 47.
00:26 7.
00:28 We can now find for the item b what is the sample standard deviation, because we need this to compute the confidence interval.
00:38 So this sample standard deviation will be the sum for all the observations that we have, which is 15.
00:44 And we need to take each value that we have, subtract the mean, compute the square of this difference for each observation, then sum these squares, and divide this by the number of derivations minus 1, 14, and then take the square root.
01:02 So this will give us that the s here is 02022.
01:10 So with this information, we can find the confidence interval as the sample mean, plus and minus a t value times the sample standard deviation divided by the square root of n.
01:25 So this t value here is because the sample size is small and we also don't know the value.
01:32 Of the standard deviation in the population.
01:36 So we can say that is unknown.
01:39 So that's why that we have to compute using the sample.
01:42 So because of this, we are going to use the t student distribution here.
01:49 And this t student distribution has degrees of freedom equals to the sample size minus 1, which is 14.
01:56 And for a 98 % confidence interval, the area left to this number is 0 .99.
02:03 So if you use the t -table with these degrees of freedom, you're going to get that this value here is 2 .62 .45...
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