Find the 95% Confidence Interval for the true population mean for the amount of soda served.

## 8.3 A Confidence Interval for A Population Proportion

114.

Insurance companies are interested in knowing the population percent of drivers who always buckle up before riding in a car.

- When designing a study to determine this population proportion, what is the minimum number you would need to survey to be 95% confident that the population proportion is estimated to within 0.03?
- If it were later determined that it was important to be more than 95% confident and a new survey was commissioned, how would that affect the minimum number you would need to survey? Why?

115.

Suppose that the insurance companies did do a survey. They randomly surveyed 400 drivers and found that 320 claimed they always buckle up. We are interested in the population proportion of drivers who claim they always buckle up.

- \(x\) = __________
- \(n\) = __________
- \(p^{\prime}\) = __________

- Define the random variables \(X\) and \(P^{\prime}\), in words.
- Which distribution should you use for this problem? Explain your choice.
- Construct a 95% confidence interval for the population proportion who claim they always buckle up.
- State the confidence interval.
- Sketch the graph.

- If this survey were done by telephone, list three difficulties the companies might have in obtaining random results.

**116**.

According to a recent survey of 1,200 people, 61% feel that the president is doing an acceptable job. We are interested in the population proportion of people who feel the president is doing an acceptable job.

- Define the random variables \(X\) and \(P^{\prime}\) in words.
- Which distribution should you use for this problem? Explain your choice.
- Construct a 90% confidence interval for the population proportion of people who feel the president is doing an acceptable job.
- State the confidence interval.
- Sketch the graph.

**117**.

An article regarding interracial dating and marriage recently appeared in the Washington Post. Of the 1,709 randomly selected adults, 315 identified themselves as Latinos, 323 identified themselves as blacks, 254 identified themselves as Asians, and 779 identified themselves as whites. In this survey, 86% of blacks said that they would welcome a white person into their families. Among Asians, 77% would welcome a white person into their families, 71% would welcome a Latino, and 66% would welcome a black person.

- We are interested in finding the 95% confidence interval for the percent of all black adults who would welcome a white person into their families. Define the random variables \(X\) and \(P^{\prime}\), in words.
- Which distribution should you use for this problem? Explain your choice.
- Construct a 95% confidence interval.
- State the confidence interval.
- Sketch the graph.

**118**.

Refer to the information in Table \(\PageIndex{5}\) shows the total receipts from individuals for a random selection of 40 House candidates rounded to the nearest $100. The standard deviation for this data to the nearest hundred is \(\sigma\) = $909,200.

$3,600 | $1,243,900 | $10,900 | $385,200 | $581,500 |

$7,400 | $2,900 | $400 | $3,714,500 | $632,500 |

$391,000 | $467,400 | $56,800 | $5,800 | $405,200 |

$733,200 | $8,000 | $468,700 | $75,200 | $41,000 |

$13,300 | $9,500 | $953,800 | $1,113,500 | $1,109,300 |

$353,900 | $986,100 | $88,600 | $378,200 | $13,200 |

$3,800 | $745,100 | $5,800 | $3,072,100 | $1,626,700 |

$512,900 | $2,309,200 | $6,600 | $202,400 | $15,800 |

Table \(\PageIndex{5}\)

- Find the point estimate for the population mean.
- Using 95% confidence, calculate the error bound.
- Create a 95% confidence interval for the mean total individual contributions.
- Interpret the confidence interval in the context of the problem.

**137**.

The American Community Survey (ACS), part of the United States Census Bureau, conducts a yearly census similar to the one taken every ten years, but with a smaller percentage of participants. The most recent survey estimates with 90% confidence that the mean household income in the U.S. falls between $69,720 and $69,922. Find the point estimate for mean U.S. household income and the error bound for mean U.S. household income.

**138**.

The average height of young adult males has a normal distribution with standard deviation of 2.5 inches. You want to estimate the mean height of students at your college or university to within one inch with 93% confidence. How many male students must you measure?

**139**.

If the confidence interval is change to a higher probability, would this cause a lower, or a higher, minimum sample size?

**140**.

If the tolerance is reduced by half, how would this affect the minimum sample size?

**141**.

If the value of \(p\) is reduced, would this necessarily reduce the sample size needed?

**142**.

Is it acceptable to use a higher sample size than the one calculated by \(\frac{z^{2} p q}{e^{2}}\)?

**143**.

A company has been running an assembly line with 97.42%% of the products made being acceptable. Then, a critical piece broke down. After the repairs the decision was made to see if the number of defective products made was still close enough to the long standing production quality. Samples of 500 pieces were selected at random, and the defective rate was found to be 0.025%.

- Is this sample size adequate to claim the company is checking within the 90% confidence interval?
- The 95% confidence interval?