ST 361 Normal Distribution



ST361: Ch 5.5 + Ch 5.6 Sampling Distribution

Topics:

I. What is Sampling Distribution?

II. Sampling Distribution of a Sample Mean [pic]

(a) X ~ Normal Distribution

(b) X ~ Non-normal Distribution

III. Central Limit Theorem

IV. Sampling Distribution of the Sample Proportion p

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I. Sampling Distribution

• Population vs. Sample:

• A Parameter is_____________________________________________

• A Statistic is _________________________________________________

• The observed value of statistic depends on the particular sample; hence it ________ from sample to sample. Such variability is called ____________________________

• The probability distribution of the statistics is called _________________________

Ex1. A neighborhood has 5 houses A, B, C, D and E. They respectively have 3, 2, 5, 3, and 4 bedrooms. We randomly draw 3 houses at a time and calculate the sample statistics median and mean. What is the sampling distribution of the sample median? What is the sampling distribution of the sample mean?

▪ Population =

▪ Variable of interest =

▪ Sample =

|Houses drawn in the sample |# of bedrooms |Sample median |Sample mean |

|ABC |3,2,5 |3 |10/3=3.3 |

|ABD | | | |

|ABE |3,2,4 |3 |9/3 = 3 |

|ACD |3,5,3 |3 |11/3 = 3.7 |

|ACE |3,5,4 | | |

|ADE |3,3,4 | |10/3 = 3.3 |

|BCD | | | |

|BCE |2,5,4 |4 |11/3 |

|BDE |2,3,4 |3 |9/3 = 3 |

|CDE |5,3,4 | | |

[pic]

II. Sampling Distribution of a Sample Mean [pic]

Let [pic] be the sample mean of a random sample [pic] from a population mean [pic] and SD [pic]. (That is, [pic].) We want to know the sampling distribution of [pic].

❖ If X ~ Normal (mean=[pic], SD=[pic]). Then [pic], the mean of a random sample of n observations

• follows a _____________________, with mean [pic] and standard deviation [pic].

• [pic]= ______________, and[pic] = ____________

• [pic] is also called standard error (SE) of [pic], or Standard error of the mean

Ex 2. Thousands of boxes contain nuts. The weights are normally distributed with mean [pic]=1 lb and SD [pic]=0.01 lb. We inspect 4 boxes and get their weights [pic]. The sample mean is [pic]

(a) What is the sampling distribution of[pic]? Mean and SE of[pic]?

(b) What is the probability that [pic] lies between 0.99 and 1.01 lb?

❖ X ~ any non-normal distribution with mean=[pic], SD=[pic]. The sampling distribution of [pic] based on samples of size n is

(a) If n is small (i.e., _______________ ), then

• Distribution:

• Mean [pic] and SE [pic]:

(b) If n is large (i.e., ________________ ), then

• Distribution:

• Mean [pic] and SE[pic]:

❖ These results follow from Central Limit Theorem (CLT)

III. Central Limit Theorem

Assume X follows an arbitrary distribution with mean [pic] and SD[pic].

When sample size is sufficiently large (i.e., n[pic]30), the sample distribution of [pic] always follows normal distribution with mean [pic] and SE [pic]

• Usually the ____________________ a distribution is, the __________ the sample size will need to ensure normality of [pic]

Ex3. Let X be the number of major defects for each new automobile tested. Suppose the number of such defects for a certain model is with mean[pic]=3.2 and SD [pic]=2.4. A sample of 100 new cars is collected.

(a) What is the sampling distribution of [pic]based on samples of size 100? What is its center and what is the SE of[pic]?

(b) What is the probability that the sample average number of major defects exceeds 4?

❖ Comments:

• If [pic] be the sample mean of a random sample [pic] from a population mean [pic] and SD [pic], then regardless of the sample size n and the distribution of X,

[pic]

• The variation of sample means is ________([pic]) than variation of the original data

• As sample size n increases, [pic] (the SE of [pic]) ____________, and the shape of the sampling distribution becomes __________________ . This implies higher probability around its mean [pic].

[pic]

Ex4. The heights of college age students (denoted by X) are known to have mean [pic]=115 and SD [pic]=30.

(a) What is the sampling distribution of[pic], the average height of 36 college age students? What are the mean and SE of the sampling distribution of[pic]?

(b) What is the sampling distribution of[pic]based on samples of 9 college age students? What are the mean and SE of the sampling distribution of [pic]?

(c) Assume that we were told that the heights of college age students are normally distributed. What is the sampling distribution of [pic]based on samples of 9 college age students? What are the mean and SE?

IV. Sampling Distribution of a Sample Proportion p

Ex. Consider a basket containing 100 balls with 2 colors: Red and White. The proportion of Red balls is denoted by [pic] (and is not known). Assume 20 balls were randomly picked from the basket with replacement, and 14 balls out of the 20 balls were red.

1) In the sample, what is the proportion of red balls?

2) We refer such quantity, [pic] as _________________________and denote it by_____.

(Note that ____________________________________________) Our question of interests: what is the distribution of the sample proportion [pic]?

Thoughts: we can think a r.v such that X = 1 if “red” and X=0 if “not red”.

Then [pic]can be view as ________________________________.

That is, [pic]is ______________________.

Thus by ___________________________, [pic]~______________if [pic]large. (However, different criteria for “large [pic]” are needed here.)

Sampling Distribution of p

(a) If _large n (i.e.,____________________________), then the sample proportion [pic] has

• A ________________________ ( by ___________________________ )

• Mean (denoted by [pic]) ’ [pic], and SE (denoted by [pic]) [pic]

(b) If _small n (i.e.,_____________________________), then the sample proportion [pic] has

• ________________________________

• Mean (denoted by [pic]) ’ ______, and SE (denoted by [pic]) = ____________

Ex5. In the population, the proportion of defectives [pic] =12%.

a) What is the sampling distribution of [pic] based on 100 observations? What is the mean? What is the standard error?

b) What is the probability that [pic] ................
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