Quantitative Methods - Calculating Confidence Intervals

When population variance (σ2) is known, the z-statistic can be used to calculate a reliability factor. Relative to the t-distribution, it will result in tighter confidence intervals and more reliable estimates of mean and standard deviation. Z-values are based on the standard normal distribution.

For establishing confidence intervals when the population variance is known, the interval is constructed with this formula:

 

Formula 2.34

For alpha of 5% (i.e. a 95% confidence interval), the reliability factor (Zα/2) is 1.96, but for a CFA exam problem, it is usually sufficient to round to an even 2 to solve the problem. (Remember that z-value at 95% confidence is 2, as tables for z-values are sometimes not provided!) Given a sample size of 16, a sample mean of 20 and population standard deviation of 25, a 95% confidence interval would be 20 + 2*(25/(16)1/2) = 20 + 2*(25/4) = 20 + 12.5. In short, for this sample size and for these sample statistics, we would be 95% confident that the actual population mean would fall in a range from 7.5 to 32.5.

Suppose that this 7.5-to-32.5 range was deemed too broad for our purposes. Reducing the confidence interval is accomplished in two ways: (1) increasing sample size, and (2) decreasing our allowable level of confidence.

1. Increasing sample size from 16 to 100 - Our 95% confidence is now equal to 20 + 2*(25/(100)1/2) = 20 + 2*(25/10) = 20 + 5. In other words, increasing the sample size to 100 narrows the 95% confidence range: min 15 to max 25.

2. Using 90% confidence - Our interval is now equal to 20 + 1.65*(25/(100)1/2) = 20 + 1.65*(25/10) = 20 + 4.125. In other words, decreasing the percentage confidence to 90% reduces the range: min 15.875 to max 24.125.

When population variance is unknown, we will need to use the t-distribution to establish confidence intervals. The t-statistic is more conservative; that is, it results in broader intervals. Assume the following sample statistics: sample size = 16, sample mean = 20, sample standard deviation = 25.

To use the t-distribution, we must first calculate degrees of freedom, which for sample size 16 is equal to n - 1 = 15. Using an alpha of 5% (95% confidence interval), our confidence interval is 20 + (2.131) * (25/161/2), which gives a range minimum of 6.68 and a range maximum of 33.32.

As before, we can reduce this range with (1) larger samples and/or (2) reducing allowable degree of confidence:

1. Increase sample size from 16 to 100 - The range is now equal to 20 + 2 * (25/10) à minimum 15 and maximum 25 (for large sample sizes the t-distribution is sufficiently close to the z-value that it becomes an acceptable alternative).

2. Reduce confidence from 95% to 90% - The range is now equal to 20 + 1.65 * (25/10) à minimum 15.875 and maximum 24.125.

Large Sample Size
In our earlier discussion on the central limit theorem, we stated that large samples will tend to be normally distributed even when the underlying population is non-normal. Moreover, at sufficiently large samples, where there are enough degrees of freedom, the z and t statistics will provide approximately the same reliability factor so we can default to the standard normal distribution and the z-statistic. The structure for the confidence interval is similar to our previous examples.

For a 95% confidence interval, if sample size = 100, sample standard deviation = 10 and our point estimate is 15, the confidence interval is 15 + 2* (10/1001/2) or 15 + 2. We are 95% confident that the population mean will fall between 13 and 17.

Suppose we wanted to construct a 99% confidence interval. Reliability factor now becomes 2.58 and we have 15 + 2.58*(10/1001/2) or 15 + 2.58, or a minimum of 12.42 and a maximum of 17.58.

The table below summarizes the statistics used in constructing confidence intervals, given various situations:

 

Distribution Population Variance Sample Size Appropriate Statistic
Normal Known Small z
Normal Known Large z
Normal Unknown Small t
Normal Unknown Large t or z
Non-Normal Known Small unavailable
Non-Normal Known Large z
Non-Normal Unknown Small unavailable
Non-Normal Unknown Large t or z


Exam Tips and Tricks
While these calculations don\'t seem difficult, it\'s true that this material seems at times to run together, particularly if a CFA candidate has never used it or hasn\'t studied it in some time. While not likely to be a major point of emphasis, expect at least a few questions on confidence intervals and in particular, a case study that will test basic knowledge of definitions, or that will compare/contrast the two statistics presented (t-distribution and z-value) to make sure you know which is useful in a given application. More than anything, the idea is to introduce confidence intervals and how they are constructed as a prerequisite for hypothesis testing

Hypothesis Testing
Related Articles
  1. Personal Finance

    Top 10 Most Valuable Sports Teams in 2015

    Cleats, pads and profits: we take a look at the top 10 most valuable sports teams in the world.
  2. Professionals

    Chinese Slowdown Affects Iron Ore Market

    The Chinese economy's ongoing slowdown is having a major impact on iron ore demand.
  3. Personal Finance

    Invest in Costco? First Understand Its Balance Sheet

    A strong balance sheet sets a company apart and boosts investor confidence. How healthy is Costco based on an analysis of its balance sheets from the last two years?
  4. Investing Basics

    Brokers and RIAs: One and the Same?

    Brokers and registered investment advisors have some key differences. Here's what you need to know.
  5. Professionals

    DCF Vs. Comparables: Which One To Use

    DCF and Comparables models are widely used in equity valuation. We explain the pros and cons of each method.
  6. Professionals

    How To Make Money Using Tobin's Q Ratio

    Although it seems simple, Tobin's Q Ratio is more complex than it appears. We explore some of its main strengths and weaknesses.
  7. Taxes

    3 Secrets You Didn't Know About Estate Planning

    Every advisor and saver needs to know these three estate planning secrets.
  8. Professionals

    Cash Flow Is King: How to Keep it Running

    Why is cash flow so important, and what steps can a business take to improve it?
  9. Entrepreneurship

    10 Ways to Nurse Cash Flow in Healthcare

    Running a business in healthcare? You might want to rethink cash flow management practices.
  10. Professionals

    How to Help Clients with Cash Flow Issues

    Sometimes your spending gets out of hand or income has a hiccup. Here's how financial advisors can help clients who have cash flow issues.
RELATED TERMS
  1. Personal Financial Advisor

    Professionals who help individuals manage their finances by providing ...
  2. CFA Institute

    Formerly known as the Association for Investment Management and ...
  3. Chartered Financial Analyst - CFA

    A professional designation given by the CFA Institute (formerly ...
  4. Security Analyst

    A financial professional who studies various industries and companies, ...
RELATED FAQS
  1. What are the differences between a Chartered Financial Analyst (CFA) and a Certified ...

    The differences between a Chartered Financial Analyst (CFA) and a Certified Financial Planner (CFP) are many, but comes down ... Read Full Answer >>
  2. How do I become a Chartered Financial Analyst (CFA)?

    According to the CFA Institute, a person who holds a CFA charter is not a chartered financial analyst. The CFA Institute ... Read Full Answer >>
  3. What types of positions might a Chartered Financial Analyst (CFA) hold?

    The types of positions that a Chartered Financial Analyst (CFA) is likely to hold include any position that deals with large ... Read Full Answer >>
  4. Who benefits the most from prepaid expenses?

    Prepaid expenses benefit both businesses and individuals. Prepaid expenses are the types of expenses that are bought or paid ... Read Full Answer >>
  5. If I am looking to get an Investment Banking job. What education do employers prefer? ...

    If you are looking specifically for an investment banking position, an MBA may be marginally preferable over the CFA. The ... Read Full Answer >>
  6. Can I still pass the CFA Level I if I do poorly in the ethics section?

    You may still pass the Chartered Financial Analysis (CFA) Level I even if you fare poorly in the ethics section, but don't ... Read Full Answer >>
Trading Center
×

You are using adblocking software

Want access to all of Investopedia? Add us to your “whitelist”
so you'll never miss a feature!