According to a survey, only 15% of customers who visited the web site of a major retail store made a purchase. Random samples of size 50 are selected.
a) What proportion of the samples will have less than 15% of customers who will make a purchase after visiting the web site?
a)
______________
b) What proportion of the samples will have between 20% and 30% of customers who will make a purchase after visiting the web site?
b)
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c) The standard deviation of all the sample proportions of customers who will make a purchase after visiting the web site is ________.
c)
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d) The average of all the sample proportions of customers who will make a purchase after visiting the web site is ________.
d)
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e) 90% of the samples will have more than what percentage of customers who will make a purchase after visiting the web site?
e)
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To answer these questions, we can use the concept of sampling distribution and the properties of proportions.
a) To find the proportion of samples with less than 15% of customers making a purchase, we need to find the cumulative probability of the sample proportion being less than 15%.
To do this, we can use the formula for the standard error of a proportion:
SE = sqrt((p_hat * (1 - p_hat)) / n)
Where:
- p_hat is the estimated proportion (15% or 0.15 in this case)
- n is the sample size (50 in this case)
Once we have the standard error, we can convert it into a z-score using the standard normal distribution table. Since we are interested in the proportion being less than 15%, we need to find the area to the left of 15% on the standard normal distribution curve.
Using the z-score, we can find the cumulative probability using the standard normal distribution table. The result will be the proportion of samples with less than 15% of customers making a purchase.
b) To find the proportion of samples with between 20% and 30% of customers making a purchase, we follow a similar approach. Find the z-scores for both 20% and 30% using the standard error formula. Then find the area between these two z-scores using the standard normal distribution table. The result will be the proportion of samples falling within this range.
c) The standard deviation of all the sample proportions can be calculated using the formula:
SD = sqrt((p * (1 - p)) / n)
Where:
- p is the true proportion in the population (15% or 0.15 in this case)
- n is the sample size (50 in this case)
d) The average of all the sample proportions is equal to the true proportion in the population. In this case, it is 15% or 0.15.
e) To find the percentage of customers that 90% of the samples will have more than, we need to find the z-score that corresponds to the cumulative probability of 90% on the standard normal distribution table. Once we have the z-score, we can convert it back to a proportion using the standard error formula. Finally, subtract this proportion from 1 to get the percentage of customers that is higher.
Please note that to get precise numerical values for these answers, you would need to perform the calculations using a standard normal distribution table or statistical software.