The mean volume for a bottle of cologne is 4 ounces and the standard deviation is 0.22 ounces. A random sample of 121 bottles is taken. What is the probability of obtaining a sample mean within 0.04 ounces of the population mean?

a. 0.9545
b. 0.0363
c. 0.3085
d. 0.5000

Answers

Answer 1

Answer:

The standard error of the mean is 0.02

Step-by-step explanation:

Here, we want to calculate the standard error of the mean.

Mathematically;

standard error of mean = SD/ √n

where SD is the standard deviation and n is the number of samples

From the question, SD = 0.22 and n = 121

Inserting this into the equation

Standard error of the mean = 0.22/√121

= 0.22/11 = 0.02

Answer 2

The probability of obtaining a sample mean is "0.9545".

According to the question,

Mean,

[tex]\mu = 4[/tex]

Standard deviation,

[tex]\sigma = 0.22[/tex]

Random sample,

[tex]n = 121[/tex]

By Central limit theorem,

→ [tex]\bar {x} \sim (4, \frac{0.22}{\sqrt{121} } )[/tex]

     [tex]= 0.2[/tex]

Now,

→ [tex]P(|\bar{x} - \mu| \ \alpha \ 0.04) = P(-0.04 \ \alpha \ \bar{x} - \mu \ \alpha \ 0.04)[/tex]

                               [tex]= P(\frac{-0.04}{0.02} \ \alpha \ z \ \alpha \ \frac{0.04}{0.02} )[/tex]

                               [tex]= P(-2 \ \alpha \ z \ \alpha \ 2)[/tex]

                               [tex]= 0.9545[/tex]

Thus the above response i.e., "option a" is appropriate.

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