In a grocery store, a clerk prepares bags of spices of the...

In a grocery store, a clerk prepares bags of spices of the same size by manually pouring the spice. The weight in grams (X) of each bag is a random variable with mean $\mu = 100$ g and standard deviation $\sigma = 5$ g. Each bag is then sold for $0.25 per gram. a) Determine the variance of Y, where Y represents the price (in dollars) of a randomly selected bag of spices b) Determine the variance of Z, where Z represents the total price (in dollars) of 10 randomly selected bags of spices. The weights of the bags are assumed to be independent. c) You combine the contents of 15 randomly selected bags of spices, which you then edistribute into 10 bags of identical weight. Determine the expectation of U, where U represents the weight of one of the reconstituted bags.

Transcript

00:01 A quality characteristics of interest for tea bag filling process is the weight of the tea in the individual bags.

00:08 If the bags are underfilled, two problems arise.

00:11 First, the customers may not be able to brew the tea to be as strong as they wish.

00:17 And second, the company may be in violation of the truth in laboring laws.

00:22 For this product, the label weight on the package indicates that on average, there are 5 .5 grams of tea in a bag.

00:29 If the mean amount of tea in the bag exceeds the label weight, the company is giving away product.

00:36 Getting an exact amount of tea in a bag is problematic because of variation in the temperature and humidity inside the factory.

00:44 Differences in the density of the tea and extremely fast filling operation of the machine, approximately 170 bags per minute.

00:53 So with this, i have, so the file tea bags contains these weights in grams of a sample of 50 tea bags.

01:02 So here is the sample of the 50 tea bags.

01:07 So it wants you to compute the mean, median, first quartile and third quartile.

01:13 So first thing i'm gonna do is i'm gonna find the mean and the mean remember is the average.

01:18 So to get the mean, you have to total up all the numbers.

01:24 So the sum of values is, i'm just gonna make this.

01:34 So sum of the values and i'm gonna total up the sum here is 274 .46.

01:46 And to get the mean, i'm going to take my sum and i'm going to divide it by 50 because that's how many numbers that there are.

01:56 And my sum or my mean is 5 .4892.

02:02 And so that's my mean.

02:04 My median is the number that's in the middle.

02:07 So that's in between the 25th and 26th number.

02:10 So i'm just gonna highlight these.

02:12 So 5 .5 is in the middle and because it's in between 5 .5 and 5 .5, so the median is 5 .5.

02:23 And then it wants the first quartile.

02:26 So first quartile and the first quartile is in between the 12th and the 13th number.

02:35 So that is going to be, so if i take this number and i'm going to take the 12th number, add it to the 13th number, and then i'm going to, let me just put this in here.

02:52 I'm gonna divide that by two and i'm gonna get 5 .405.

03:01 And then i need to, so i'm just gonna highlight these.

03:05 So then i'm going to get between the, so 42, 43, so 25 or 26 plus 12.

03:25 So that's between the, so one, two, three, four, five, six, seven, eight, nine, 10, 11, 12.

03:32 So between the 37th and 38th number.

03:35 So if i highlight those, so i have one, two, three, four, five, six, seven, eight, nine, 10, 11, 12.

03:45 And then one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13 and 25.

03:54 Ooh, it's 5 .41, sorry, the 13th number.

03:58 And hold on, let me just, no fail.

04:01 So it's 5 .41, i messed up on that, i apologize, 5 .41.

04:05 And then the third quartile is 5 .57...

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