Chapter 6: Confidence Intervals

Section 2

Confidence Intervals for the Mean (*σ* Unknown)

*Constructing a Confidence Interval for a Population Proportion* (p. 321)

*In Words:* *(1):* Identify the sample statistics *n* and *x*. *(2):* Find the point estimate *p̂*. *(3):* Verify that the sampling distribution *p̂* can be approximated by a normal distribution. *(4):* Find the critical value *z*∨*c* that corresponds to the given level of confidence *c*. (*Use Table 4 in Appendix B.*) *(5):* Find the margin of error *Ε*. *(6):* Find the left and right endpoints and form the confidence interval. *In Symbols:* *(2):* *p̂* = *x* / *n* *(3):* *np̂* ≥ 5, *nq̂* ≥ 5 *(5):* *Ε* = *z*∨*c* × √((*p̂*×*q̂*)/(*n*) *(6):* Left endpoint: *p̂* - *Ε* Right endpoint: *p̂* + *Ε* Interval: *p̂* - *Ε* < *p* < *p̂* + *Ε*

*Constructing a Confidence Interval for a Population Mean ("σ" Unknown)* (p. 312)

*In Words:* *(1.)* Verify that *σ* is not known, the sample is random, and either the population is normally distributed or *n* ≥ 30. *(2.)* Find the sample statistics *n*, *x̄*, and *s*. *(3.)* Identify the degrees of freedom, the level of confidence *c*, and the critical value *t*∨*c*. *(4.)* Find the margin of error *Ε*. *5.)* Find the left and right endpoints and form the confidence interval. *In Symbols:* *(2.)* *x̄* = ((*∑x*)/*n*), *s* = √(((*∑(x - x̄)²*) / (*n* - 1)) *(3.)* d.f. = *n* - 1 (*Use Table 5 in Appendix B.*) *(4.)* *Ε* = *t*∨*c* × ((*s*) / (√*n*)) *(5.)* Left endpoint: *x̄* - *Ε* Right endpoint: *x̄* + *Ε* Interval: *x̄* - *Ε* < *μ* < *x̄* + *Ε*

*Constructing a Confidence Interval for a Population Mean ("σ" Known)* (p. 301)

*In Words:* *1.)* Verify that *s* is known, the sample is random, and either the population is normally distributed or *n* ≥ 30. *2.)* Find the sample statistics *n* and *x̄*. *3.)* Find the critical value *z*∨*c* that corresponds to the given level of confidence. (*Use Table 4 in Appendix B.*) *4.)* Find the margin of error *Ε*. *5.)* Find the left and right endpoints and form the confidence interval. *In Symbols:* *2.)* *x̄* = (*∑x*) / *n* *4.)* *Ε* = *z*∨*c* × *(σ/√n)* *5.)* Left endpoint: *x̄* - *Ε* Right endpoint: *x̄* + *Ε* Interval: *x̄* - *Ε* < *μ* < *x̄* + *Ε*

*c*-confidence interval for a population proportion *p*

*p̂* - *Ε* < *p* < *p̂* + *Ε* where *Ε* = *z*∨*c* × √((*p̂*×*q̂*)/(*n*). ~ (Margin of error for *p*) The probability that the confidence interval contains *p* is *c*, assuming that the estimation process is repeated a large number of times. (p. 321)

point estimate

A single value estimate for a population parameter. The most unbiased point estimate of the population mean *μ* is the sample *x̄*. (p. 298)

interval estimate

An interval, or range of values, used to estimate a population parameter. (p. 299)

Section 3

Confidence Intervals for Population Proportions

Section 1

Confidence Intervals for the Mean (*σ* Known)

*(Y.T.I.) Exercise 3 (p. 301):* Construct the confidence interval for the population mean *μ*. (Round answers to *two* decimal places.) *c = 0.95, x̄ = 4.8, σ = 0.7*, and *n = 55* A 95% confidence interval for *μ* is (*__(1)__*, *__(2)__*).

Correct Answer(s): *(1):* *4.61* *(2):* *4.99*

*(Y.T.I.) Exercise 2 (p. 300):* Find the margin of error for the given values of​ *c*, *σ*​, and *n*. (Round answer to *three* decimal places.) *c = 0.90, σ = 3.2, n = 49* *Level of Confidence* = *z*∨*c* *90%* = *1.645* *95%* = *1.96* *99%* = *2.575* *E* = *____*

Correct Answer: *0.752*

*(Y.T.I.) Exercise 1 (p. 311):* Find the critical value *t*∨*c* for the confidence level *c* = 0.98 and sample size *n* = 26. (Round answer to the *nearest thousandth* (*three* decimal places).) *t*∨*c* = *____*

Correct Answer: *2.485*

*(Y.T.I.) Exercise 6 (p. 304):* A cheese processing company wants to estimate the mean cholesterol content of all​ one-ounce servings of a type of cheese. The estimate must be within 0.74 milligram of the population mean. *Part 1 (a):* Determine the minimum sample size required to construct a 95% confidence interval for the population mean. (Round answer to the *nearest whole number* (*zero* decimal places).) Assume the population standard deviation is 3.02 milligrams. *Part 2 (b):* The sample mean is 31 milligrams. Using the minimum sample size with a 95% level of​ confidence, does it seem likely that the population mean could be within 3% of the sample​ mean? Within 0.3% of the sample​ mean? Explain. (Round answers to *two* decimal places.) *Part 1 (a):* The minimum sample size required to construct a 95% confidence interval is *__* servings. *Part 2 (b):* The 95% confidence interval is (*__(1)__*, *__(2)__*). It *__(3)__* likely that the population mean could be within 3% of the sample mean because the interval formed by the values 3% away from the sample mean *_____(4)_____* the confidence interval. It *__________(5)__________* seem likely that the population mean could be within 0.3% of the sample mean because the interval formed by the values 0.3% away from the sample mean *__________(6)__________* the confidence interval.

Correct Answers: *Part 1 (a):* *64* *Part 2 (b):* *(1):* *30.26* *(2):* *31.74* *(3):* *seems* *(4):* *entirely contains* *(5):* *does not seem* *(6):* *overlaps but does not entirely contain*

*(Y.T.I.) Exercise 3 (p. 313):* In a random sample of ten cell​ phones, the mean full retail price was $535.50 and the standard deviation was $150.00. Assume the population is normally distributed and use the​ *t* - distribution to find the margin of error and construct a 95% confidence interval for the population mean μ. Interpret the results. *Part 1:* Identify the margin of error. (Round answer to *one* decimal place.) *__(1)__* *___(2)___* *Part 2:* Construct a 95% confidence interval for the population mean. (Round answers to *one* decimal place.) (*__(1)__*, *__(2)__*) *Part 3:* Interpret the results. Select the correct choice below and fill in the answer box to complete your choice. (Type answer(s) as either *integers* or *decimals*, but *DO NOT ROUND*.) A.) With *__%* ​confidence, it can be said that most cell phones in the population have full retail prices​ (in dollars) that are between the​ interval's endpoints. B.) *__%* of all random samples of ten people from the population of cell phones will have a mean full retail price​ (in dollars) that is between the​ interval's endpoints. C.) It can be said that *__%* of the population of cell phones have full retail prices​ (in dollars) that are between the​ interval's endpoints. D.) With *__%* confidence, it can be said that the population mean full retail price of cell phones​ (in dollars) is between the​ interval's endpoints.

Correct Answers: *Part 1:* *(1):* *107.2* *(2):* *dollars* *Part 2:* *(1):* *428.3* *(2):* *642.7* *Part 3:* D.) With *95%* confidence, it can be said that the population mean full retail price of cell phones​ (in dollars) is between the​ interval's endpoints.

*(Y.T.I.) Exercise 5 (p. 303):* You are given the sample mean and the population standard deviation. Use this information to construct the 90% and 95% confidence intervals for the population mean (round all answers to *two* decimal places). Interpret the results and compare the widths of the confidence intervals. If​ convenient, use technology to construct the confidence intervals. *A random sample of 60 home theater systems has a mean price of $118.00. Assume the population standard deviation is $19.90.* *Part 1:* Construct a 90% confidence interval for the population mean. The 90% confidence interval is ​(*___(1)___*, *___(2)___*). *Part 2:* Construct a 95% confidence interval for the population mean. The 95% confidence interval is (*___(1)___*, *___(2)___*). *Part 3:* Interpret the results. Choose the correct answer below. A.) With 90% confidence, it can be said that the population mean price lies in the first interval. With 95% confidence, it can be said that the population mean price lies in the second interval. The 95% confidence interval is wider than the 90%. B.) With 90% confidence, it can be said that the sample mean price lies in the first interval. With 95% confidence, it can be said that the sample mean price lies in the second interval. The 95% confidence interval is wider than the 90%. C.) With 90% confidence, it can be said that the population mean price lies in the first interval. With 95% confidence, it can be said that the population mean price lies in the second interval. The 95% confidence interval is narrower than the 90%.

Correct Answers: *Part 1:* *(1):* *113.77* *(2):* *122.23* *Part 2:* *(1):* *112.96* *(2):* *123.04* *Part 3:* A.) With 90% confidence, it can be said that the population mean price lies in the first interval. With 95% confidence, it can be said that the population mean price lies in the second interval. The 95% confidence interval is wider than the 90%.

*(Y.T.I.) Exercise 2 (p. 312):* In a random sample of 29 people, the mean commute time to work was 30.9 minutes and the standard deviation was 7.3 minutes. Assume the population is normally distributed and use a​ *t* - distribution to construct a 90% confidence interval for the population mean *μ*. What is the margin of error of *μ*​? Interpret the results. (Round all answers to *one* decimal place.) *Part 1:* The confidence interval for the population mean *μ* is (*__(1)__*, *__(2)__*). *Part 2:* The margin of error of *μ* is *__*. *Part 3:* Interpret the results. A.) With 90% confidence, it can be said that the population mean commute time is between the bounds of the confidence interval. B.) With 90% confidence, it can be said that the commute time is between the bounds of the confidence interval. C.) It can be said that 90% of people have a commute time between the bounds of the confidence interval. D.) If a large sample of people are taken, approximately 90% of them will have commute times between the bounds of the confidence interval.

Correct Answers: *Part 1:* *(1):* *28.7* *(2):* *33.1* *Part 2:* *2.2* *Part 3:* A.) With 90% confidence, it can be said that the population mean commute time is between the bounds of the confidence interval.

*(Y.T.I.) Exercise 1 (p. 320):* Let p be the population proportion for the following condition. Find the point estimates for p and q. (Round both answers to *three* decimal places.) *In a survey of 1,937 adults from country A, 437 said that they were not confident that the food they eat in country A is safe.* *Part 1:* The point estimate for p, *p̂*, is *____*. *Part 2:* The point estimate for q, *q̂*, is *____*.

Correct Answers: *Part 1:* *0.226* *Part 2:* *0.774*

*Finding a Minimum Sample Size to Estimate "μ"* (p. 304)

Given a *c*-confidence level and a margin of error *Ε*, the minimum sample size *n* needed to estimate the population mean *μ* is: *n* = ((*z*∨*c* × *σ*)/*Ε*)². If *n* is not a whole number, then round *n* up to the next whole number. Also, when *σ* is unknown, you can estimate it using *s*, provided you have a preliminary random sample with at least 30 members.

*Finding a Minimum Sample Size to Estimate "p"*

Given a *c*-confidence level and a margin of error *Ε*, the minimum sample size *n* needed to estimate the population proportion *p* is: *n* = (*p̂* × *q̂*) × (((*z*∨*c*)/(*Ε*))²) If *n* is not a whole number, then round *n* up to the next whole number. Also, note that this formula assumes that you have preliminary estimates of *p̂* and *q̂*. If not, use *p̂* = 0.5 and *q̂* = 0.5. (p. 324)

margin or error (*Ε*) (p. 300)

Given a level of confidence *c*, it is the greatest possible distance between the point estimate and the value of the parameter it is estimating. For a population mean *μ* where *σ* is known, the margin of error is: *E* = *z*∨*c* × *σ∨x̄* = *z*∨*c* × *(σ/√n)* *(Margin of error for μ (σ known))* when these conditions are met. *1.)* The sample is random. *2.)* At least one of the following is true: The population is normally distributed or *n* ≥ 30. (Recall from the Central Limit Theorem that when *n* ≥ 30, the sampling distribution of sample means approximates a normal distribution.)

*t* - distribution (p. 310)

If the distribution of a random variable x is approximately normal, then *t* = (*x̄* - *μ*) / (*s* / (*√n*)) follows a *t* - distribution. Critical values of *t* are denoted by *t*∨*c*. Here are several properties of the *t* - distribution. *1.)* The mean, median, and mode of the *t* - distribution are equal to 0. *2.)* The *t* - distribution is bell-shaped and symmetric about the mean. *3.)* The total area under the *t* - distribution curve is equal to 1. *4.)* The tails in the *t* - distribution are "thicker" than those in the standard normal distribution. *5.)* The standard deviation of the *t* - distribution varies with the sample size, but it is greater than 1. *6.)* The *t* - distribution is a family of curves, each determined by a parameter called the "degrees of freedom". The *degrees of freedom* (sometimes abbreviated as *d.f.*) are the number of free choices left after a sample statistic such as x is calculated. When you use a *t* - distribution to estimate a population mean, the degrees of freedom are equal to one less than the sample size. *Degrees of freedom:* *d.f.* = *n* - 1 *7.)* As the degrees of freedom increase, the *t* - distribution approaches the standard normal distribution, as shown in the figure. For 30 or more degrees of freedom, the *t* - distribution is close to the standard normal distribution. (Since I don't have Quizlet+, I can't insert the image of the actual standard normal distribution; ergo, I pasted [some of] the graph info.) *d.f.* = *2* *d.f.* = *5*

sampling error

The difference between the point estimate and the actual parameter value. (p. 300)

unbiased estimator

The mean of its sampling distribution is equal to the true value of the parameter being estimated. (p. 298)

degrees of freedom

The number of free choices left after a sample statistic such as *x̄* is calculated. (Sometimes abbreviated as *d.f.*) (p. 310)

point estimate for *p*

The population proportion of success, is given by the proportion of successes in a sample and is denoted by: *p̂* = *x* / *n* (*Sample proportion*) where *x* is the number of successes in the sample and *n* is the sample size; the point estimate for the population proportion of failures is *q̂* = 1 - *p̂*; the symbols *p̂* and *q̂* are read as "*p* hat" and "*q* hat". (p. 320)

population proportion

The probability of success in a single trial of a binomial experiment. (P. 320)

*c*-confidence interval for a population mean *μ* (p. 301)

The probability that the confidence interval contains *μ* is *c*, assuming that the estimation process is repeated a large number of times. *x̄* - *Ε* < *μ* < *x̄* + *Ε*

level of confidence

The probability that the interval estimate contains The population parameter, assuming that the estimation process is repeated a large number of times. (p. 299)

critical values

Values that separate sample statistics that are probable from sample statistics that are improbable or unusual. (p. 299)