Stat 321 – Lecture



Topic: Normal Distribution

Recap:

• Continuous random variable: takes all possible values in an interval

• Probability density function (pdf): f(x) such that P(ax) = P(X>x)

• Cumulative distribution function: F(x) = P(X Normal specifying the mean, SD, and input constant.

(n) See how this prediction matches our exam data in C1:

MTB> let c13= (c1>31.8 & c1 tally c13

(o) Do a similar tally with the Arby’s sandwiches (use the sample mean and standard deviation of C5 to determine the two values [pic] and [pic] and then use the let command to determine how many data values fall in the interval). What percentage of the observations fall “within one standard deviation of the mean”?

(p) What percentage of the observations fall “within two standard deviations of the mean”?

While these calculations have been approximations, the general empirical rule is that with data that roughly follow a normal distribution:

• Roughly 68% of the observations fall within 1 standard deviation of the mean

• Roughly 95% of the observations fall within 2 standard deviations of the mean

• Roughly 99.7% of the observations fall within 3 standard deviations of the mean

These can be verified for the theoretical normal distribution using Table A.3 or Minitab.

Normal Approximation to Binomial

Now examine the data in C8, let X = number of boys in a family of four children.

(q) Is X a discrete or continuous random variable? Explain.

(r) Can X follow an exact normal distribution? Explain.

(s) What distribution does X follow exactly? (Give the name and the parameter values.)

Often we can approximate a binomial distribution with the normal distribution. In general, the rule of thumb is that this approximation applies well when np> 10 and n(1-p)>10. When this rule of thumb is satisfied, we can approximate a binomial(n, p) distribution with a Normal(μ=np, σ=[pic]) distribution.

(t) Simulate 500 observations from a Binomial distribution with n=50 and p=.5

MTB> random 500 c15;

SUBC> bino 50 .5.

Examine a histogram of this distribution. Does it appear approximately normal? Does it satisfy the above conditions for the normal approximation to the binomial to be valid?

(u) Simulate 500 observations from a Binomial distribution with n=50 and p=.05. Examine a histogram of this distribution. Does it appear approximately normal? If not, what shape does it have? Does it satisfy the above conditions for the normal approximation to be valid?

(v) Return to the scenario in (t). Suppose we wanted to calculated P(X< 20).

• Use the binomial distribution to find this probability exactly:

MTB> cdf 20;

SUBC> bino 50 .5.

• Use the normal distribution with mean =50(.5) and SD=[pic] to approximate this probability:

MTB> cdf 20;

SUBC> normal 25 3.54.

To see why this approximation is not very accurate, consider P(X=20).

(w) What is P(X=20) in the binomial distribution? What is P(X=20) in the normal distribution?

(x) To make sure X=20 is “included” in the probability, we will make a “continuity correction” and actually calculate P(X cdf –1.90;

SUBC> norm 0 1.

or Calc>Prob Dist > Normal

specifying μ, σ, x

Using Minitab:

MTB> cdf 70;

SUBC> norm 51.1 19.29.

Using Minitab:

MTB> invcdf .90;

SUBC> norm 51.1 19.29.

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