Let X represent the amount of gasoline (gallons) purchased...

Let $X$ represent the amount of gasoline (gallons) purchased by a randomly selected customer at a gas station. Suppose that the mean and standard deviation of $X$ are 11.5 and 4.0 , respectively. (a) In a sample of 50 randomly selected customers, what is the approximate probability that the sample mean amount purchased is at least 12 gallons? (b) In a sample of 50 randomly selected customers, what is the approximate probability that the total amount of gasoline purchased is at least 600 gallons? (c) What is the approximate value of the 95 th percentile for the total amount purchased by 50 randomly selected customers?

Let $X$ represent the amount of gasoline (gallons) purchased by a randomly selected customer at a
gas station. Suppose that the mean and standard deviation of $X$ are 11.5 and 4.0 , respectively.
(a) In a sample of 50 randomly selected customers, what is the approximate probability that the sample mean amount purchased is at least 12 gallons?
(b) In a sample of 50 randomly selected customers, what is the approximate probability that the total amount of gasoline purchased is at least 600 gallons?
(c) What is the approximate value of the 95 th percentile for the total amount purchased by 50 randomly selected customers?

Key Concepts

Distribution of the Sum

When taking the sum of i.i.d. random variables, the total also follows a normal distribution (under the conditions of the CLT) with a mean equal to n times the population mean and a standard deviation equal to the population standard deviation multiplied by the square root of the sample size (??n). This property is used to compute probabilities and percentiles for the total amount purchased.

Percentile Calculation

Percentile calculation involves determining the value below which a given percentage of observations in a distribution falls. In the context of a normal distribution, this is typically done by identifying the corresponding z-score and then converting it to the original scale of the variable by using the distribution's mean and standard deviation. This is particularly useful for finding the 95th percentile in the total sum distribution of purchases.

Central Limit Theorem

The Central Limit Theorem (CLT) states that, given a sufficiently large sample size, the distribution of the sample mean of independent and identically distributed (i.i.d.) random variables will be approximately normal regardless of the underlying distribution of the data. This concept is essential for approximating probabilities and percentiles for the sample mean or total sum when dealing with real-world data such as gasoline purchases.

Standard Error

The Standard Error is the standard deviation of the sampling distribution of the sample mean. It is calculated by dividing the population standard deviation by the square root of the sample size (?/?n). This measure quantifies the variability of the sample mean from the actual population mean and is central to determining the probability of the sample mean exceeding a specific value.

Transcript

00:01 We're given a random variable x, representing the amount of gasoline and gallons purchased by a randomly selected customer at a gas station.

00:11 We're told that the mean and standard deviation of x are 11 .5 and 4 .0.

00:17 In part a, we're asked to find, in a sample of 50 randomly selected customers, the approximate probability that the sample mean amount purchased is at least 12 gallons.

00:29 So we have that x bar is the sample average amount of gas purchased by 50 customers.

01:09 Now we know by the central limit theorem, since 50 is greater than 30, this applies, we have that x bar is approximately normal.

01:33 With expected value of x bar being the same as the average of the population, which we're given is 11 .5, and the standard deviation.

01:50 Of x bar is the standard deviation of the population sigma over the square root of the number of samples n and we're given that sigma is 4 .0 and we have that n is 50 and so we have 4 over root 50 and therefore the probability that x bar is going to be at least 12 this this is the same approximately as 1 minus phi of 12 minus the mean 10 .5, sorry, 11 .5, i mean.

03:10 In standard deviation, 4 .0 over root 50.

03:19 This is approximately 1 minus phi of .88.

03:31 And using a calculator or a table, we find that this is equal to approximately .1 94 .1894.

03:46 Next, in part b, we're given a sample of 50 randomly selected customers and we're asked to find the approximate probability that the total amount of gasoline purchased is at least 600 gallons.

04:00 So now we're dealing with the total amount of gasoline purchased...

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