Question

It seems that driving with a pet in the car is the third worst driving distraction, behind talking on the phone and texting. According to an American Automobile Association study, \(80 \%\) of drivers admit to driving with a pet in the car, and of those, \(20 \%\) allow their dogs to sit on their laps. \({ }^{13}\) Suppose that you randomly select a sample of 100 drivers who have admitted to driving with a pet in their car. a. What is the probability that \(25 \%\) or more of the drivers allow their dogs to sit on their laps? b. What is the probability that \(10 \%\) or fewer of the drivers allow their dogs to sit on their laps? c. Would it be unusual to find that \(35 \%\) of the drivers allow their dogs to sit on their laps?

Short answer

Is it unusual to find 35% of the drivers allowing their dogs on their laps? Answer: To find the probability that 25% or more drivers allow their dogs on their laps, calculate the sum of the binomial distribution probabilities for values from 25 to 100. Similarly, for 10% or fewer drivers, calculate the sum for values from 0 to 10. If the probability for 35% of the drivers (35 drivers) is less than 0.05, it is considered unusual.

Step-by-step solution

Find the range of interest

We want to find the probability that 25% or more of the drivers allow their dogs on their laps. Since our sample size is 100, this means we want the probability that 25 or more drivers belong to this category.

Calculate the probabilities

We will apply the binomial distribution formula for each value in the range 25 to 100: $$ P(X \geq 25) = \sum_{k=25}^{100} {100\choose k} (.20)^k (.80)^{100-k} $$ b. Probability that 10% or fewer allow their dogs on their laps

Find the range of interest

We want to find the probability that 10% or fewer of the drivers allow their dogs on their laps. Since our sample size is 100, this means we want the probability that 10 or fewer drivers belong to this category.

Calculate the probabilities

We will apply the binomial distribution formula for each value in the range 0 to 10: $$ P(X \leq 10) = \sum_{k=0}^{10} {100\choose k} (.20)^k (.80)^{100-k} $$ c. Unusual to find 35% of the drivers allow their dogs on their laps

Determine if value is unusual

35% of 100 drivers means 35 drivers. If the probability of having 35 drivers who allow their dogs on their laps is very low (less than 5% or 0.05), then we consider it to be unusual.

Calculate the probability

We will apply the binomial distribution formula for the value of 35: $$ P(X = 35) = {100\choose 35} (.20)^{35} (.80)^{65} $$

Compare to criterion for "unusual" values

If the calculated probability is less than 0.05, then we consider it unusual for 35% of the drivers to allow their dogs on their laps.

Key concepts

Probability and Its Importance

Probability is the likelihood or chance of an event occurring. When we talk about probability in the context of a binomial distribution, as in our exercise involving drivers and pets, we focus on events that can be categorized into two outcomes, such as success or failure.
In our problem, letting a dog sit on the driver's lap is considered the "success" event, while not letting the dog sit there is the "failure" event. By assigning probabilities to these events, we can calculate how likely it is for a certain number of drivers to behave in a particular way.

• The probability function can help us predict outcomes based on given probabilities for a sample size, like 100 drivers in this case.
• This involves the application of a formula to determine the likelihood of events, such as having 25% or more, or 10% or fewer drivers allowing dogs on their laps.

Understanding probability is crucial for making informed decisions and predictions based on data.

The Role of Statistical Analysis

Statistical analysis involves collecting, analyzing, interpreting, presenting, and organizing data. It's a tool that enables us to make sense of a large set of numbers and to determine meaningful patterns or probabilities.
In the context of our problem, statistical analysis allows us to use the binomial distribution to calculate probabilities of known outcomes, such as the chance that 10% or fewer drivers let their dogs sit on their laps.

• We apply statistical methods to assess the distribution of the sample data to find out how likely certain numbers of successes are.
• This involves using well-known probability distribution formulas, like the binomial distribution, to model our scenarios mathematically.

Statistical analysis transforms raw numbers into understandable insights, letting us infer the characteristics or trends of a larger population from a sample.

Understanding Sample Size in Studies

Sample size is a term used to define the number of subjects included in a survey or experiment. In statistical studies, the size of your sample can greatly affect the reliability of your conclusions.
In our problem, the sample size is 100 drivers. This size allows us to apply the binomial model effectively. However, when the sample size is either too small or too large, it can lead to inaccurate or misleading conclusions.

• The larger the sample size, the more reliable the statistical findings are likely to be; as it is expected to better mirror the population.
• A sample that is too small might not represent the overall population accurately, leading to incorrect predictions or assumptions.

Understanding the role of sample size is essential to properly interpret the results of a statistical analysis and to extend those insights to a more comprehensive group.