Question

As mentioned in the chapter, opinion-polling organizations contact their respondents by sampling random telephone numbers. Although interviewers now can reach about \(76 \%\) of U.S. households, the percentage of those contacted who agree to cooperate with the survey has fallen from \(58 \%\) in 1997 to only \(38 \%\) in 2003 (Pew Research Center for the People and the Press). Each household, of course, is independent of the others. a) What is the probability that the next household on the list will be contacted but will refuse to cooperate? b) What is the probability (in 2003 ) of failing to contact a household or of contacting the household but not getting them to agree to the interview? c) Show another way to calculate the probability in part b.

Short answer

a) 0.4712; b) 0.7112; c) 0.7112 (alternative calculation).

Step-by-step solution

Probability of Contacting Household

The probability of contacting a household is given as 76% or 0.76.

Probability of Refusing to Cooperate

The probability of a household that has been contacted refusing to cooperate is given for 2003 as 62% (100% - 38% cooperation rate).Thus, \[P(\text{Refuse to Cooperate}|\text{Contacted}) = 0.62\]

Probability of Contact and Refusal

To find the probability that a household is contacted and refuses to cooperate, multiply the probability of contacting with the probability of refusal:\[P(\text{Contacted and Refuse}) = P(\text{Contacted}) \times P(\text{Refuse to Cooperate}|\text{Contacted}) = 0.76 \times 0.62 = 0.4712\]

Probability of Not Contacted or Contacted and Refusal (Part b)

For part b, use the rule of total probability. The events "not contacted" or "contacted and refuse" are mutually exclusive.\[P(\text{Not Contacted or Contacted and Refuse}) = P(\text{Not Contacted}) + P(\text{Contacted and Refuse})\]\[P(\text{Not Contacted}) = 1 - P(\text{Contacted}) = 1 - 0.76 = 0.24\]Thus, \[P(\text{Not Contacted or Contacted and Refuse}) = 0.24 + 0.4712 = 0.7112\]

Alternative Calculation (Part c)

Another method to calculate the probability is to find the complementary probability of households that are contacted and agree (which is 0.76 \times 0.38 = 0.2888) and then subtract from 1.\[P(\text{Not Contacted or Contacted and Refuse}) = 1 - P(\text{Contacted and Agree}) = 1 - 0.2888 = 0.7112\]

Key concepts

Opinion Polling

Opinion polling is an essential tool used by researchers to gauge public sentiment. It involves collecting responses from a sample group that represents a larger population.
Polls help in understanding the general opinion on various topics, from political preferences to consumer habits.

Here are some important points about opinion polling:

• **Sample Size**: A good poll needs a sample that's representative of the entire population. The larger the sample, generally, the more accurate the results.
• **Random Sampling**: To avoid bias, participants are randomly chosen. This ensures each member of the population has an equal chance of being selected.
• **Margin of Error**: This is a statistic expressing the amount of random sampling error in a poll's results. A smaller margin indicates more accurate results.
• **Confidence Level**: This is how confident pollsters are that the survey results reflect the true opinion of the population. It's usually set at 95%.

Understanding polling is crucial, especially in making informed decisions based on statistical evidence.

Telephone Surveys

Telephone surveys are a common method used in opinion polling. They involve contacting people over the phone to gather their opinions or views.
This method is traditionally preferred because it reaches a wide audience quickly and efficiently.

When conducting telephone surveys, consider the following:

• **Reachability**: While a significant portion of the population may be contacted via phone, not everyone has access or is willing to participate.
• **Response Rates**: These have been declining over the years. In the given example, the cooperation rate dropped significantly from 58% in 1997 to 38% in 2003.
• **Cost Efficiency**: Telephone surveys tend to be more cost-effective than face-to-face interviews, but the challenge remains in ensuring respondents' willingness.

Telephone surveys provide valuable data, but researchers should be mindful of the reach and cooperation challenges.

Statistical Independence

Statistical independence is a vital concept in probability. It refers to scenarios where the occurrence of one event does not affect the probability of another.
In the context of telephone surveys, each household's decision to participate is independent of others.

Here’s how you can frame the idea of statistical independence in these surveys:

• **Independent Events**: Every household's decision is independent. The probability of being contacted and the refusal to cooperate are distinct events.
• **Calculating Probabilities**: You can calculate scenarios such as the probability of a household refusing to cooperate using independent probabilities. For example, the probability of a household being contacted and then refusing is calculated separately.
• **Application**: For part (c) of the exercise, the concept helps use an alternative approach by calculating probabilities through complementary methods.

This understanding is pivotal for any statistical analysis, allowing for more accurate computations and predictions.