According to a librarian, the distribution of library fines paid by the students has a mean of RM6 and a standard deviation of RM4. Find the probability that the sample mean will be above 6.3 when the sample size is 35.a.0.95b.0.67c.0.33d.0.48Clear my choice
Question
According to a librarian, the distribution of library fines paid by the students has a mean of RM6 and a standard deviation of RM4. Find the probability that the sample mean will be above 6.3 when the sample size is 35.a.0.95b.0.67c.0.33d.0.48Clear my choice
Solution
To solve this problem, we need to use the Central Limit Theorem which states that if you have a population with mean μ and standard deviation σ and take sufficiently large random samples from the population with replacement, then the distribution of the sample means will be approximately normally distributed.
Step 1: Identify the parameters The population mean (μ) is RM6 and the population standard deviation (σ) is RM4. The sample size (n) is 35.
Step 2: Calculate the standard deviation of the sample mean The standard deviation of the sample mean (σx̄) is calculated as σ/√n. Substituting the given values, we get σx̄ = 4/√35 = 0.676.
Step 3: Calculate the z-score The z-score is calculated as (x̄ - μ) / σx̄. We want to find the probability that the sample mean is above RM6.3, so x̄ = 6.3. Substituting these values, we get z = (6.3 - 6) / 0.676 = 0.44.
Step 4: Find the probability The z-score of 0.44 corresponds to a probability of 0.67 (from z-table or using a standard normal distribution calculator). However, since we want the probability that the sample mean is above RM6.3, we need to subtract this value from 1 (because the total probability under the normal curve is 1). So, the required probability is 1 - 0.67 = 0.33.
So, the answer is c. 0.33.
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