It is generally believed that electrical problems affect about 14% of new cars. An automobile mechanic conducts diagnostic tests on 128 new cars on the lot.
a. Describe the sampling distribution for sample proportion by naming the model and telling its mean and standard deviation. Justify your answer.
b. What is the probability that in this group over 18% of the new cars will be found to have electrical problems?
ahosooo
osv
a. The sampling distribution for sample proportion in this scenario can be approximated by the normal distribution. This is justified by two main reasons:
1. The sample size is reasonably large: The sample size of 128 cars is considered large enough to satisfy the conditions for using the normal distribution approximation. According to the Central Limit Theorem, as long as the sample size is sufficiently large, the sampling distribution of the sample proportion will be approximately normal.
2. The population from which the sample is taken is unknown, but it is reasonable to assume that the proportion of cars with electrical problems follows a normal distribution: Although we do not have specific information about the population, it is generally reasonable to assume that the proportion of cars with electrical problems in new cars follows a normal distribution.
The mean of the sampling distribution of the sample proportion can be estimated as the population proportion, which is 14% or 0.14. The standard deviation of the sampling distribution can be calculated using the formula:
Standard deviation (σ) = sqrt(p * (1 - p) / n)
where p is the population proportion (0.14) and n is the sample size (128).
b. To find the probability that over 18% of the new cars will be found to have electrical problems, we need to calculate the probability of obtaining a sample proportion greater than 18% from the sampling distribution.
First, we standardize the threshold value (18%) using the formula:
Z = (x - μ) / σ
where x is the threshold value (0.18), μ is the mean of the sampling distribution (0.14), and σ is the standard deviation of the sampling distribution.
Next, we use a standard normal distribution table or a statistical software to determine the probability of obtaining a value greater than the standardized threshold Z.
By looking up the Z-score (calculated in the previous step) in the standard normal distribution table or using a statistical software, we can determine the probability associated with the Z-score. This probability represents the likelihood of observing a sample proportion greater than 18% in this scenario.