A firm claims that only 10% of its accounts receivables are over 30 days past due. The bank that supplies working capital or interim loans to the firm is suspect of the figure and accordingly takes a random of 100 accounts of the firm. Find probability that the sample proportion, p, will be: a) between 9% and 10%. b) at least 12%.
To find the probability in each case, we need to calculate the sample proportion, p, and then use the normal distribution to find the probabilities.
a) To find the probability that the sample proportion, p, is between 9% and 10%, we need to calculate the z-scores for each percentage and use the standard normal distribution table.
Step 1: Calculate the sample proportion, p
Since the firm claims that only 10% of the accounts receivables are over 30 days past due, we use this as the population proportion, P. The sample proportion, p, is then the proportion of accounts in the sample that are over 30 days past due.
p = Number of accounts over 30 days past due / Sample size
p = 10 / 100 = 0.1
Step 2: Calculate the z-scores
To find the z-score for each percentage, we use the formula:
z = (p - P) / sqrt(P * (1 - P) / n)
For 9%:
z1 = (0.09 - 0.10) / sqrt(0.10 * (1 - 0.10) / 100)
For 10%:
z2 = (0.10 - 0.10) / sqrt(0.10 * (1 - 0.10) / 100)
Step 3: Use the standard normal distribution table
Look up the probabilities corresponding to the z-scores calculated in Step 2 using the standard normal distribution table. Subtract the probability corresponding to z1 from the probability corresponding to z2 to get the final probability.
b) To find the probability that the sample proportion, p, is at least 12%, we need to calculate the z-score for 12% and then use the standard normal distribution table.
Step 1: Calculate the sample proportion, p
p = 12 / 100 = 0.12
Step 2: Calculate the z-score
z = (p - P) / sqrt(P * (1 - P) / n)
For 12%:
z = (0.12 - 0.10) / sqrt(0.10 * (1 - 0.10) / 100)
Step 3: Use the standard normal distribution table
Look up the probability corresponding to the z-score calculated in Step 2 using the standard normal distribution table. Subtract this probability from 1 to get the final probability.