A firm claims that only 10% of it's accounts receivables are over 30 days past due. The bank that supplies working capital or interim laons to the fimrs is suspect of the figure and accordingly takes a random samole of 100 accounts of the firm. Find the probability that the sample proportiom'p'will be:
a)between 9% and 10%.
b)at least 12%.
To solve this problem, we can use the concept of sampling distributions and the normal distribution. Here are the steps to calculate the probabilities:
Step 1: Finding the mean and standard deviation of the sample proportion:
To find the mean (μ) of the sample proportion, we use the proportion of accounts receivables that are over 30 days past due, which is 10% or 0.10.
μ = 0.10
For the standard deviation (σ) of the sample proportion, we use the formula:
σ = sqrt((p * (1-p)) / n)
where p is the population proportion (0.10) and n is the sample size (100).
σ = sqrt((0.10 * (1-0.10)) / 100) = 0.030
Step 2: Standardizing the values:
To answer the first part of the question, we need to find the probability of the sample proportion being between 9% and 10%. We can do this by standardizing the values using the formula:
Z = (x - μ) / σ
where x is the value we want to standardize.
For 9%:
Z1 = (0.09 - 0.10) / 0.030 = -0.033
For 10%:
Z2 = (0.10 - 0.10) / 0.030 = 0
Step 3: Finding the probabilities:
Using a standard normal distribution table or a calculator, we can find the probabilities associated with the standardized values.
a) Probability between 9% and 10%:
P(0.09 ≤ p ≤ 0.10) = P(Z1 ≤ Z ≤ Z2)
= P(-0.033 ≤ Z ≤ 0)
= P(Z ≤ 0) - P(Z ≤ -0.033)
You can find these probabilities from a standard normal distribution table or use a calculator.
b) Probability of at least 12%:
To find the probability that the sample proportion is at least 12%, we need to standardize the value of 12% and find the probability associated with it.
Z3 = (0.12 - 0.10) / 0.030
P(p ≥ 0.12) = P(Z ≥ Z3)
= P(Z ≥ (0.12 - 0.10) / 0.030)
Again, you can find this probability from a standard normal distribution table or use a calculator.
By following these steps, you can calculate the probabilities for the given situations.