Sampling Distribution (Chapter 7)

Why is it important to check if the 10% condition is met before calculating probabilities involving x-bar? (a) to reduce the variability of the sampling distribution of x-bar (b) to ensure that the distribution of x-bar is approx. normal (c) to ensure that we can generalize the results to a larger population (d) to ensure that x-bar will be a an unbiased estimator of mu. (e) to ensure that the observations in the sample are close to independent

(e)

Normal approximation

(large counts) when the sample size n is large, the sampling distribution of ^p is close to a Normal distribution. When both are greater or equal to 10

Suppose a large candy machine has 45% orange candies. Sampling 1: Mean=0.449 Std dev=0.105; sample size 25, number of samples 400 Sampling 2: Mean 0.446, Std dev=0.070, sample size 50, number of samples 400 Which is more surprising: getting a sample of 25 candies in which 32% are orange or getting a sample of 50 in which 32% are orange?

A sample of 50, because we expect to be closer to p=0.45 in larger samples.

A bottling company uses a filling machine to fill plastic bottles with cola. The bottles are supposed to contain 300 millimeters. In fact, the contents vary according to a Normal distribution with mean of 298 ml and std dev of 3 ml. What is the probably that a randomly selected bottle contains less than 295 ml?

There is a 0.1587 probability that a randomly selected bottle contains less than 295 ml.

Suppose a large candy machine has 45% orange candies. Mean=0.449 Std dev=0.105; sample size 25, number of samples 400 Would you be surprised if a sample of 25 candies from the machine contained 8 orange candies? How about 5 orange candies? Explain.

We would not be surprised to find 8 (32%) orange candies because values this small happened fairly often in the simulation. However, there were few samples in which there were few samples in which there were 5 (20%) or fewer orange candies. So getting 5 orange candies would be surprising.

Statistic

a number that describes some characteristic of a sample

Parameter

a number that describes some characteristic of the population

Unbiased Estimator

a statistic used for estimating a parameter is unbiased if the mean of its sampling distribution is equal to the true value of the parameter being estimated

Biased Estimator

a statistic used to estimate a parameter is biased if the mean of its sampling distribution is not equal to the true value of the parameter being estimated

Variability of a Statistic

described by the spread of its sampling distribution, determined mainly by the size of the random sample

Sampling Distribution of ^P

describes how the sample proportion varies in all possible samples from the population.

Population Distribution

gives the values of the variable for all individuals in the population

mean of the sampling distribution of ^p

is equal to the population proportion (^P is an unbiased estimator of ^p)

Central Limit Theorem (CLT)

As the size n of a simple random sample increases, the shape of the sampling distribution of x̄ tends toward being normally distributed.(n>_30)

Sampling Distribution of a Statistic

the distribution of values taken by the statistic in all possible samples of the same size from the same population

Identify the population, the parameter, the sample, and the statistic in each setting: A random sample of 1000 people who signed a card saying they intended to quit smoking were contacted 9 months later. It turned out that 210 (21%) of the sampled individuals had not smoked over the past 6 months.

Population: All people who signed a card saying that they intend to quit smoking Parameter: the proportion of the population who actually quit smoking. Sample: a random sample of 1000 people who signed the cards. Statistic: the proportion of the sample who actually quit smoking; p-hat=0.21

Identify the population, the parameter, the sample, and the statistic in each setting: Tom is cooking a large turkey breast for a holiday meal. He wants to be sure that the turkey is safe to eat, which requires a minimum internal temperature of 165 degrees Fahrenheit. Tom uses a thermometer to measure the temperature of the turkey meat at four randomly chosen points. The minimum reading in the sample is 170 degrees.

Population: all the turkey meat. Parameter: minimum temperature in all of the turkey meat. Sample: four randomly chosen locations in the turkey. Statistic: minimum temperature in the sample of four locations. sample minimum =170

Standard deviation of the sampling distribution of ^p

as long as the 10% condition is satisfied: n is greater or equal to 1/10N

Distribution of Sample Data

shows the values of the variable for the individuals in the sample