# Stats 300 - Chapter Seven

## Unlock all answers in this set

Unlock answers (30)
question
sample proportion
answer
The sample proportion is the best point estimate of the population proportion.
question
point estimate
answer
single value (or point) used to approximate a population parameter.
question
confidence interval (or interval estimate)
answer
range (or an interval) of values used to estimate the true value of a population parameter. A confidence interval is sometimes abbreviated as CI.
question
confidence level, degree of confidence, or the confidence coefficient.
answer
is the probability 1 - Î± (often expressed as the equivalent percentage value) that the confidence interval actually does contain the population parameter, assuming that the estimation process is repeated a large number of times.
question
Margin of Error for Proportions
answer
E = z(Î±/2) * ( [pÌ‚ * qÌ‚] / n ) ^(1/2)
question
Confidence Interval for Estimating a Population Proportion p
answer
pÌ‚ - E < pÌ‚ < pÌ‚ + E where E = z(Î±/2) * ( [pÌ‚ * qÌ‚] / n ) ^(1/2)
question
Confidence Interval for Estimating a Population Proportion p notation
answer
pÌ‚ - E < pÌ‚ < pÌ‚ + E pÌ‚ +/- E (pÌ‚ - E, pÌ‚ + E)
question
Procedure for Constructing a Confidence Interval for p
answer
1.Verify that the required assumptions are satisfied. (The sample is a simple random sample, the conditions for the binomial distribution are satisfied, and the normal distribution can be used to approximate the distribution of sample proportions because np >= 5, and nq >= 5 are both satisfied.) 2. Refer to Table A-2 and find the critical value z(Î±/2) that corresponds to the desired confidence level. 3. Evaluate the margin of error 4. Using the value of the calculated margin of error, E and the value of the sample proportion, p, find the values of p - E and p + E. Substitute those values in the general format for the confidence interval: pÌ‚ - E < pÌ‚ < pÌ‚ + E 5. Round the resulting confidence interval limits to three significant digits.
question
Sample Size for Estimating Proportion p
answer
When an estimate of pÌ‚ is known n = (z(Î±/2)^2 * pÌ‚ * qÌ‚) / E^2 When no estimate of p is known: n = (z(Î±/2)^2 * 0.25) / E^2
question
Round-Off Rule for Determining Sample Size
answer
If the computed sample size n is not a whole number, round the value of n up to the next larger whole number.
question
Finding the Point Estimate and E from a Confidence Interval
answer
Point estimate of pÌ‚ = (upper confidence limit + lower confidence limit) / 2 Margin of error E = (upper confidence limit - lower confidence limit) / 2
question
Point Estimate of the Population Mean
answer
The sample mean xbar is the best point estimate of the population mean Âµ.
question
Confidence Interval for Estimating a Population Mean (with Ïƒ Known)
answer
1. The sample is a simple random sample. (All samples of the same size have an equal chance of being selected.) 2. The value of the population standard deviation Ïƒ is known. 3. Either or both of these conditions is satisfied: The population is normally distributed or n > 30.
question
Confidence Interval for Estimating a Population Mean (with Ïƒ Known)
answer
xbar - E < Î¼ < xbar + E or xbar +/- E or (xbar - E, xbar + e) where E = z(Î±/2) * ( Ïƒ / (n)^(1/2) )
question
confidence interval limits
answer
xbar - E, xbar + E
question
Sample Mean
answer
1. For all populations, the sample mean x is an unbiased estimator of the population mean xbar, meaning that the distribution of sample means tends to center about the value of the population mean Î¼. 2. For many populations, the distribution of sample means x tends to be more consistent (with less variation) than the distributions of other sample statistics.
question
Procedure for Constructing a Confidence Interval for Âµ (with Known Ïƒ)
answer
1. Verify that the requirements are satisfied. 2. Refer to Table A-2 or use technology to find the critical value z(Î±/2) that corresponds to the desired confidence level 3. Evaluate the margin of error E = z(Î±/2) * ( Ïƒ / (n)^(1/2) ) 4. Find the values of xbar - E and xbar + E. Substitute those values in the general format of the confidence interval 5. Round using the confidence intervals round-off rules.
question
Round-Off Rule for Confidence Intervals Used to Estimate Âµ
answer
When using the original set of data, round the confidence interval limits to one more decimal place than used in original set of data. When the original set of data is unknown and only the summary statistics (n, x, s) are used, round the confidence interval limits to the same number of decimal places used for the sample mean.
question
Finding a Sample Size for Estimating a Population Mean
answer
n = ( [z(Î±/2) * Ïƒ] / E)^2
question
Round-Off Rule for Sample Size n
answer
If the computed sample size n is not a whole number, round the value of n up to the next larger whole number.
question
Sample Mean
answer
The sample mean is the best point estimate of the population mean.
question
Student t Distribution
answer
If the distribution of a population is essentially normal, then the distribution of t = (xbar - Î¼) / [ s / n^(1/2) ]
question
degrees of freedom
answer
The number of degrees of freedom for a collection of sample data is the number of sample values that can vary after certain restrictions have been imposed on all data values. The degree of freedom is often abbreviated df. degrees of freedom = n - 1 in this section
question
Procedure for Constructing a Confidence Interval for Âµ (With Ïƒ Unknown)
answer
1. Verify that the requirements are satisfied. 2. Using n - 1 degrees of freedom, refer to Table A-3 or use technology to find the critical value t(Î±/2) that corresponds to the desired confidence level. 3. Evaluate the margin of error E = t(Î±/2) â€¢ [ s / n^(1/2 ] . 4. Find the values of xbar - E and xbar + E. Substitute those values in the general format for the confidence interval: xbar - E < Î¼ < xbar + E 5. Round the resulting confidence interval limits
question
Important Properties of the Student t Distribution
answer
1. The Student t distribution is different for different sample sizes (see the following slide, for the cases n = 3 and n = 12). 2. The Student t distribution has the same general symmetric bell shape as the standard normal distribution but it reflects the greater variability (with wider distributions) that is expected with small samples. 3. The Student t distribution has a mean of t = 0 (just as the standard normal distribution has a mean of z = 0). 4. The standard deviation of the Student t distribution varies with the sample size and is greater than 1 (unlike the standard normal distribution, which has a Ïƒ = 1). 5. As the sample size n gets larger, the Student t distribution gets closer to the normal distribution.
question
Choosing the Appropriate Distribution
answer
Use the normal (z) distribution If Ïƒ known and normally distributed population or Ïƒ known and n > 30 Use t distribution if Ïƒ not known and normally distributed population or Ïƒ not known and n > 30 Use a nonparametric method or bootstrapping If Population is not normally distributed and n â‰¤ 30
question
Finding the Point Estimate and E from a Confidence Interval
answer
Point estimate of Âµ: xbar = (upper confidence limit + lower confidence limit) / 2 Margin of Error: E = (upper confidence limit - lower confidence limit) / 2
question
Chi-Square Distribution
answer
In a normally distributed population with variance Ïƒ^2 assume that we randomly select independent samples of size n and, for each sample, compute the sample variance s2 (which is the square of the sample standard deviation s). The sample statistic x^2 (pronounced chi-square) has a sampling distribution called the chi-square distribution.
question
Chi-Square Distribution formula
answer
X^2 = ( [ n -1 ] * s^2) / Ïƒ^2
question
Properties of the Distribution of the Chi-Square Statistic
answer
1. The chi-square distribution is not symmetric, unlike the normal and Student t distributions. As the number of degrees of freedom increases, the distribution becomes more symmetric. 2. The values of chi-square can be zero or positive, but they cannot be negative. 3. The chi-square distribution is different for each number of degrees of freedom, which is df = n - 1. As the number of degrees of freedom increases, the chi-square distribution approaches a normal distribution. In Table A-4, each critical value of X^2 corresponds to an area given in the top row of the table, and that area represents the cumulative area located to the right of the critical value.
question
Confidence Interval for Estimating a Population Standard Deviation or Variance
answer
( [n - 1] * s^2 ) / X(r)^2 < Ïƒ^2 < ( [ n - 1 ] * s^2) / X(L)^2
question
Procedure for Constructing a Confidence Interval for Ïƒ or Ïƒ^2
answer
1. Verify that the required assumptions are satisfied. 2. Using n - 1 degrees of freedom, refer to Table A-4 or use technology to find the critical values X(r)^2 and X(L)^2 that correspond to the desired confidence level/ 3. Evaluate the upper and lower confidence interval limits using this format of the confidence interval: ( [n - 1] * s^2 ) / X(r)^2 < Ïƒ^2 < ( [ n - 1 ] * s^2) / X(L)^2 4. If a confidence interval estimate of is desired, take the square root of the upper and lower confidence interval limits and change Ïƒ^2 to Ïƒ. 5. Round the resulting confidence level limits. If using the original set of data to construct a confidence interval, round the confidence interval limits to one more decimal place than is used for the original set of data. If using the sample standard deviation or variance, round the confidence interval limits to the same number of decimals places.
question
Confidence Intervals for Comparing Data Caution
answer
Confidence intervals can be used informally to compare the variation in different data sets, but the overlapping of confidence intervals should not be used for making formal and final conclusions about equality of variances or standard deviations.
question
Critical Value
answer
A critical value is the number on the borderline separating sample statistics that are likely to occur from those that are unlikely to occur.