A department store has determined that 25% of all their sales are credit sales. A random sample of 60 sales is selected. The probability that the sample proportion will be less than 0.3 is .
Question
A department store has determined that 25% of all their sales are credit sales. A random sample of 60 sales is selected. The probability that the sample proportion will be less than 0.3 is .
Solution
To solve this problem, we will use the concept of Normal Approximation to the Binomial Distribution.
Step 1: Identify the given values. The proportion of credit sales, p = 0.25 (25%) The sample size, n = 60 We want to find the probability that the sample proportion is less than 0.3.
Step 2: Calculate the mean and standard deviation. The mean (μ) of a binomial distribution is np. So, μ = np = 60 * 0.25 = 15 The standard deviation (σ) is sqrt(np(1-p)). So, σ = sqrt(60 * 0.25 * (1 - 0.25)) = sqrt(11.25) = 3.35 (approximately)
Step 3: Standardize the desired proportion. We want to find the probability that the sample proportion (p̂) is less than 0.3. In terms of the number of sales, this is 0.3 * 60 = 18 sales. To standardize this, we subtract the mean and divide by the standard deviation. This gives us a z-score. Z = (18 - 15) / 3.35 = 0.895 (approximately)
Step 4: Look up the probability associated with this z-score. Using a standard normal distribution table or a calculator, we find that the probability associated with a z-score of 0.895 is approximately 0.815.
So, the probability that the sample proportion will be less than 0.3 is approximately 0.815.
Similar Questions
A survey found that 32%of consumers from a Country A are more likely to buy stock in a company based in Country A,or shop at its stores,if it is making an effort to publicly talk about how it is becoming more sustainable.Suppose you select a sample of 200 respondents from Country A.Complete parts(a)through(d)below. d.If a sample of 800 is taken,how does this change your answers to(a)through(c)? If a sample of 800 is taken,what is the probability that in the sample fewer than 32%are more likely to buy stock in a company based in Country A,or shop at its stores,if it is making an effort to publicly talk about how it is becoming more sustainable? The probability is %， (Round to two decimal places as needed
Laptop buyers can pay their purchases in cash, or using a credit card, or a debit card. The manager observed that55% of the shoppers used credit cards, 15% paid in cash, and the rest paid with their debit cards.a. Write the sample space:b. Assign the probability for each sample space.
An on-line retailing firm conducts a study into the number of on-line purchases of footwear made during all of last year. The study wants to determine the proportion of Australians over the age of 25 who purchase footwear on-line. The managing director of the on-line retailing firm believes this proportion of Australians is 22%. A random sample of 179 Australians over the age of 25 is selected. If the managing director’s claim is true, what is the sample proportion value below which 10% of all sample proportions would exist? Express your answer correct to two decimal places.
A marketing survey is conducted in which students are to taste two different brands of soft drink.Their task is to correctly identify the brand tasted.A random sample of 200 students is taken.Assume that the students have no ability to distinguish between the two brands. b.The probability is 90%that the sample percentage is contained within what symmetrical limits of the population percentage?Identify the limits of the population percentage. The lower limit is The upper limit is (Round to four decimal places as needed.)
A manufacturing company is concerned about the number of defective items produced by their assembly line. In the past they have had 5% of their products produced defectively. They take a random sample of 35 products. What is the probability that more than 5 products in the sample are defective?
Upgrade your grade with Knowee
Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions.