  Find the probability and interpret the results. If convenient, use technology to find the probability. During a certain week the mean price of gasoline was \$2703 per gallon.  A random sample of 34 gas stations is drawn from this population. What is the probability that the mean price for the sample was between 2682 and 2722 that week? assume σ =​\$0.041Can you please explain me this math problem.  I am having so much trouble and go step by step to help me with this math problem.  Thank you.

Question

Find the probability and interpret the results. If convenient, use technology to find the probability. During a certain week the mean price of gasoline was \$2703 per gallon.  A random sample of 34 gas stations is drawn from this population. What is the probability that the mean price for the sample was between 2682 and 2722 that week? assume σ =​\$0.041

Can you please explain me this math problem.  I am having so much trouble and go step by step to help me with this math problem.  Thank you.

Step 1

Central Limit Theorem for mean:

If a random sample of size n is taken from a population having mean  and standard deviation  then, as the sample size increases, the sample mean approaches the normal distribution with mean  and standard deviation σ/ sqrt(n).

Step 2

Find the probability that the mean price for the sample was between 2,682 and 2,722:

The mean price of the gasoline is \$2,703.That is, μ = \$2,703.  Random samples of 34 (n) gas stations are drawn from the population. The population standard deviation is σ = \$0.041. The random variable X denotes the price of the gasoline. By central limit theorem for mean, the mean price of the sample follows a normal distribution with mean μ = \$2,703 and standard deviation 0.041/sqrt (34) = 0.007.Thus, the probability that the mean price for the sample was between 2,682 and 2,722 is calculated as follows:

Step 3

Interpretation:

The probability that mean price for the sample was between 2,682 and 2,722 is 1. Therefore event that means price for the sample was between 2,682 and 2,722 is a sure event. It is expected th...

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