The mean amount purchased by a typical customer at...

The mean amount purchased by a typical customer at Churchill's Grocery Store is $\$ 23.50,$ with a standard deviation of $\$ 5.00 .$ Assume the distribution of amounts purchased follows the normal distribution. For a sample of 50 customers, answer the following questions. a. What is the likelihood the sample mean is at least $\$ 25.00 ?$ b. What is the likelihood the sample mean is greater than $\$ 22.50$ but less than $\$ 25.00 ?$ c. Within what limits will $90 \%$ of the sample means occur?

The mean amount purchased by a typical customer at Churchill's Grocery Store is $\$ 23.50,$ with a standard deviation of $\$ 5.00 .$ Assume the distribution of amounts purchased follows the normal distribution. For a sample of 50 customers, answer the following questions.
a. What is the likelihood the sample mean is at least $\$ 25.00 ?$
b. What is the likelihood the sample mean is greater than $\$ 22.50$ but less than $\$ 25.00 ?$
c. Within what limits will $90 \%$ of the sample means occur?

Key Concepts

Normal Distribution

The normal distribution is a continuous probability distribution characterized by a symmetric, bell-shaped curve. It is defined by its mean and standard deviation, and many real-world phenomena approximate this distribution, making it useful for modeling and statistical inference.

Central Limit Theorem

The Central Limit Theorem states that, regardless of the original distribution of a population, the distribution of the sample mean will approach a normal distribution as the sample size increases. This theorem allows for the use of normal probability methods when dealing with sample means, even if the original data is not normally distributed.

Standard Error

The standard error is the standard deviation of a sampling distribution, specifically the distribution of the sample mean. It quantifies the amount of variability in the sample mean from sample to sample and is calculated by dividing the population standard deviation by the square root of the sample size.

Z-Score Transformation

The z-score transformation converts a sample mean to a standardized form by subtracting the population mean and dividing by the standard error. This facilitates the calculation of probabilities using the standard normal distribution since z-scores indicate how many standard deviations a value is from the mean.

Confidence Interval

A confidence interval is a range of values derived from the sample data that is believed, with a certain level of confidence, to contain the true population parameter. It provides an estimate of the uncertainty associated with a sample statistic, such as the mean.

Transcript

00:01 All right.

00:03 So we have a grocery store who's mean amount purchased.

00:10 My typical customer is 2350.

00:12 Standard deviation of five bucks.

00:15 And we're going to assume that the purchases are, the amounts are normally distributed.

00:23 And if we sample 50 customers, we're going to answer some questions about the likelihood of certain purchase amounts of samples.

00:31 So what's the likelihood of the sample? is at least 25 bucks.

00:35 So let's do this.

00:37 We need, so we're going to do our little z score calculator here, and then we'll look that value up in the table.

00:47 So we have 25.

00:48 It's a value that we're looking at, 23 .5, divided by this standard bva, or the standard error, which is sigma divided by the square root of the number of customers in the sample.

01:07 That's the z score then we're going to do a table lookup but i've got a formula that does that for me but you do the same thing if your table you get the same value 0 .98 but wait a second hold up that is not the answer you want we said at least 25 bucks this is the z score for 25 or less so more than that we can only do 1 minus it so it's 0 .01 so it's not very much now we're going to say greater than 2250 but less than 25.

01:48 So we have the 25 amount already kind of right here.

01:51 So we need to find the z score of 2250, find the area up to that point, and then subtract that from this large value here, which was it.

02:02 0 .98, something like that.

02:03 Yeah, 0 .98.

02:06 Because the way my table works or this formula works, it gives you the area up to the z score.

02:11 So the z score for 2250.

02:18 Rather than rewrite it, i'm just going to change the 25 to 2250.

02:24 That's it.

02:25 There we go.

02:26 So now we take the norm...