Soap powder labeled "Giant Size 32 Ounces." The actual weight of soap powder in such a box has a Normal distribution with a mean of 33 oz and a standard deviation of 0.7 oz. To avoid having dissatisfied customers, the company says a box of soap is considered underweight if it weighs less than 32 oz. What proportion of boxes

1) C

2) B

Step-by-step explanation:

Hello!

1.) A company produces packets of soap powder labeled “Giant-Size 32 Ounces.” The actual weight of soap powder in such a box has a Normal distribution with a mean of 33 oz and a standard deviation of 0.7 oz. To avoid having dissatisfied customers, the company says a box of soap is considered underweight if it weighs less than 32 oz. To avoid losing money, it labels the top 5% (the heaviest 5%) overweight. What proportion of boxes is underweight (i.e., weigh less than 32 oz)?

The variable of interest is

X: the weight of a soap powder box. (oz)

X~N(μ;σ²)

You need to find the proportion of boxes that weight less than 32 oz, symbolically:

P(X<32)

To calculate the proportion of soap boxes that weight less than 32 oz the best way is to use the tabulated standard normal distribution. So using this distribution Z= (X-μ)/δ~N(0;1)

By subtracting the population mean and then dividing by the standard deviation you "transform" the value of the variable X to a value of Z

P(Z<(32-33)/0.7)= P(Z<-1.428) ≅ P(Z<-1.43)

Now you can look for the corresponding value of probability in the Z-table. The value is negative so you have to use the left entry of the table. The integer and first decimal digit "-1.4*" are in the first column of the table and the second decimal digit "-.-3" is in the first row of the table, when you cross them you find the corresponding probability. (see attachment

P(Z<-1.43)= 0.0764

A) .2420

B) .9236

C) .0764

D) .7580

2.) The first quartile of any distribution is the value with 25% of observations to its left. What is the first quartile of the standard Normal distribution?

The standard normal distribution is centered in its mean μ= 0, this value corresponds to the 2nd quantile of the distribution, meaning that it divides the bottom 50% and top 50%.

The first quantile of the distribution divides the bottom 25% from the top 75%, in the case of the standard normal distribution, since it's left from the mean, this quantile is expected to be a negative number.

You can symbolize the first quartile as:

P(Z≤z₀) 0.25

Where z₀ is the value of the first quantile. Using the left entry of the distribution you have to look for the value of Z that accumulates 0.25 of probability. In this case, you have to look for the value in the body of the table and then reach for the first column and first row to see which is the corresponding Z value (See attachment)

z₀= -0.67

A) z = 0.25

B) z = - 0.67

C) cannot tell without the mean and standard deviation.

D) z = - 1.96

I hope you have a SUPER day!

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