An electric device delivers a current of [tex]$15.0 A$[/tex] for 30 seconds. How many electrons flow through it?
Answer (1)
The sample proportion P follows a normal distribution: P ∼ N ( 0.19 , 0.001304 ) .
Calculate the z-score: z = 0.001304 0.23 − 0.19 ≈ 1.108
Find the probability: 0.23) = P(Z > 1.108) \approx 0.134"> P ( P > 0.23 ) = P ( Z > 1.108 ) ≈ 0.134
The probability that more than 23% of those surveyed approve of the competitor is approximately 0.134 .
Explanation
Understand the problem and provided data First, let's identify the given information. The politician claims that the approval rating of her competitor is p = 0.19 . Maya surveys n = 118 randomly selected Californians. We want to find the probability that more than 23% of those surveyed approve of the competitor, assuming the politician's claim is true.
Determine the distribution of the sample proportion Next, we need to determine the distribution of the sample proportion P . Since the sample size n = 118 is relatively large, we can use the normal approximation to the binomial distribution. The mean of the sampling distribution of P is equal to the population proportion p , so μ = p = 0.19 . The standard deviation of the sampling distribution is given by: σ = n p ( 1 − p ) = 118 0.19 ( 1 − 0.19 ) Calculating this value: σ = 118 0.19 ( 0.81 ) = 118 0.1539 ≈ 0.001304 ≈ 0.0361 Therefore, the distribution of P is approximately normal with mean 0.19 and standard deviation 0.0361 . So, P ∼ N ( 0.19 , 0.036 1 2 ) or P ∼ N ( 0.19 , 0.001304 ) .
Calculate the z-score Now, we need to calculate the z-score for P = 0.23 . The z-score is calculated as: z = σ P − p = 0.0361 0.23 − 0.19 = 0.0361 0.04 ≈ 1.108 So, the z-score is approximately 1.108 .
Calculate the probability We want to find the probability that more than 23% of those surveyed approve of the competitor, which means we want to find 0.23)"> P ( P > 0.23 ) . This is equivalent to finding the area to the right of the z-score 1.108 in the standard normal distribution. Using a standard normal distribution table or calculator, we find that the area to the left of z = 1.108 is approximately 0.866 . Therefore, the area to the right is: 1.108) = 1 - P(Z < 1.108) = 1 - 0.866 = 0.134"> P ( Z > 1.108 ) = 1 − P ( Z < 1.108 ) = 1 − 0.866 = 0.134 So, the probability that more than 23% of those surveyed approve of the competitor is approximately 0.134 .
Final Answer In summary:
The distribution of P is approximately N ( 0.19 , 0.001304 ) .
The z-score is approximately 1.108 .
The probability that more than 23% of those surveyed approve of the competitor is approximately 0.134 .
Conclusion Therefore, the distribution is P ∼ N ( 0.19 , 0.001304 ) . The z-score is 1.108 . The probability that more than 23% of those surveyed approve of the competitor is 0.134 .
Examples
Imagine you're a marketing analyst trying to gauge the success of a new advertising campaign. You believe that 30% of customers will like the new ad. You survey 100 people and find that 35% liked it. Using this type of probability calculation, you can determine the likelihood of observing such a result if your initial belief was correct. This helps you decide whether the campaign is truly more effective than you initially thought, or if the higher percentage is just due to random chance. This is a practical application of hypothesis testing in marketing.
