STATS CHP 7

What is the equation for phat mean?

1/n (np) = p

How many times bigger must the population be than the sample?

10

In order for mean and sd of these to be true, what condition must be met?

10%

The customer care manager at a cell phone company keeps track of how long each help line caller spends on hold before speaking to a customer service representative. e finds that the distribution of wait time for al callers has a mean of 12 minutes with a SD of 5 minutes. The distribution is moderately skewed to the right. Suppose a manager takes a random sample of 10 callers and calculates their mean wait time x bar. What is the mean of sampling distribution x bar?

12 minutes

The weight of the eggs produced by a certain breed of hen is normally distributed with mean 65grams and SD 5 g. Think of cartons of such eggs as SRSs of size 12 from the population of all eggs. Calculate the probability that the mean weight of the eggs in a carton falls between 62.5 and 68.75 grams.

62.5 - 65 / 5 / root 12. <. z. <. 68.75 - 65 / 5 / root 12 -1.73 < z < 2.60 = .9535

In some cases, what can the sampling distribution of p hat be approximated by?

A normal curve

What is a statistic?

A number that describes some characteristic of a sample

What is a parameter?

A number that describes some characteristic of the population

What can you use to get a trustworthy estimate of an unknown population parameter?

A statistic that's an unbiased estimator

Why is the mean p?

Because the sample proportion p hat is an unbiased estimator of p

What happens to the distribution as n increases?

Becomes approximately normal

How do you create a sampling distribution?

By taking every one of the possible samples of size n from a population, calculating the sample proportion for each and graphing all of those values

What do our sample values not do?

Center on the population value

For the specific value of p, what happened to standard deviation as sample size increases?

Decreases

What do different random samples yield?

Different statistics

When is a statistic used to estimate a parameter an unbiased estimator?

If the mean of its sampling distribution is equal to the true value of the parameter being estomated

Why can we think of a statistic as a random variable?

It takes numerical values that describe the outcomes of the random sampling process

What give smaller spreads?

Larger samples

The weight of the eggs produced by a certain breed of hen is normally distributed with mean 65grams and SD 5 g. Did you need to know that the population distribution of egg weights was normal in order to complete parts A or B?

Yes, the calculations in both A and B assumed the normality of the underlying distribution

If x is the mean of a SRS of size n drawn from a large population with mean u and SD o, what is the mean of the distribution?

u

In a large population, 46% of the household own DVD recorders. A SRS of 10 households from this population is to be contacted and the sample proportion computed. What expression represents the probability that more than half the household sampled own a DVD recorder?

z. > .5-.46 / square root of (.46)(.54) / 100

Olive weights are classified according to a unique set of adjectives implying for great size. For example, the mean weight of olives classified as colossal is 7.7 gm. Suppose a particular company's crop of colossal olives is approximately norma distributed with a mean of 7/7gm and a SD of 0.2 gm. What represents the probability that the mean weight of a random sample of three loves is greater than 8 gm.

z. >. 8-7.7/.2/root 3

When choosing a SRS of size n and a population of size N with proportion p of successes with p hat being the sample proportion of successes, what is the SD of p hat?

square root of p(1-p) / n

What is the symbol for sample mean?

X bar

A friend has offered to play a game w you that involves flipping a coin that he has provided. Since a flip of heads will be to his advantage, you want to test the coin for fairness before you begin to play. Your friend is willing to let you flip the coin 50 times to determine if the probability of getting heads is actually .50, as it should be if the coin in fair. Assume for the moment that the coin is far. If p hat is the probability of heads in 50 flips of the coin, what are the mean and SD of the sampling distribution of p hat?

Mean = .5 SD = square root of (.5)(.5) / 50 = .071

Since p is successes over sample size, what are we doing to get the random variable p?

Multiplying the random variable X by a constant 1/n

The customer care manager at a cell phone company keeps track of how long each help line caller spends on hold before speaking to a customer service representative. e finds that the distribution of wait time for al callers has a mean of 12 minutes with a SD of 5 minutes. The distribution is moderately skewed to the right. Suppose a manager takes a random sample of 10 callers and calculates their mean wait time x bar. Do you know the approximate shape of the sampling distribution of x bar? If so, describe the shape and justify your answer. If not, explain why not.

No, the population distribution is skewed, and n = 10, which is not large enough for the central limit theorem to apply

The weight of the eggs produced by a certain breed of hen is normally distributed with mean 65grams and SD 5 g. Calculate the probability that a randomly selected egg weighs between 62.5 and 68.75 grams.

Normal cdf (62.5, 68.75, 65, 5) = .4649

When we record quantitative variables, what are we also interested in?

Other statistics such as the median, mean or SD

What is high variability?

Repeated shots are widely scattered on the target with a variety of resuktw

What are among the most common statistics?

Sample means

What does the sampling's normalcy depend on?

Sample size and population proportion

What does the standard deviation depend on?

Sample size and population proportion

If the population distribution is normal, what is also normal?

Sampling distribution of x bar

What can we use to imitate the process of taking many many samples?

Simulation

A friend has offered to play a game w you that involves flipping a coin that he has provided. Since a flip of heads will be to his advantage, you want to test the coin for fairness before you begin to play. Your friend is willing to let you flip the coin 50 times to determine if the probability of getting heads is actually .50, as it should be if the coin in fair. Assume for the moment that the coin is far. Explain why you can use the formula for the SD of p hat in this setting?

Since the population of flips of this coin is infinite, we need not to be concerned about the 10% condition for finite populations

What decreases as sample size increases?

Spread

In practice, what is it difficult to do?

Take all possible samples of size n to obtain the actual sampling distribution

If the population distribution is not normal, what does the central limit theorem state?

That the sampling distribution of x bar will be approximately normal in most cases when x>30

What does using an unbiased estimator not guarantee?

That the value of the statistic will be close to the actual parameter value

What does the central limit theorem state?

That when n is large, the sampling distribution of the sample mean x bar is approximately normal

What does an unbiased estimator ensure?

That you won't overestimate or underestimate

In order for the mean and SD to work, what condition must be met?

The 10% condition

Before performing normal calculations, what condition must be satisfied?

The Normal condition

What is the sampling distribution of a statistix?

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

Teenagers send many messages - recent polls cite medians of more than 50 per day. Consider a large population of teenagers for whom the distribution of the number of texts sent per day is strongly skewed to the right. The min is 0, q1 is 20, median is 55, q3 is 140 and max is 250. Suppose we take random samples of size 32 from this population and calculate q1 for each of the samples. Briefly explain what a dot at 35 represents.

The dot at 35 represents the first quartile of one of the 50 samples taken from this population

What allows us to answer this question: "How trustworthy is a statistic as an estimator of the parameter?"

The fact that statistics from random samples have definite sampling dostributioms

What are true despite the shape the population distribution has?

The facts about mean and SD

The customer care manager at a cell phone company keeps track of how long each help line caller spends on hold before speaking to a customer service representative. e finds that the distribution of wait time for al callers has a mean of 12 minutes with a SD of 5 minutes. The distribution is moderately skewed to the right. Suppose a manager takes a random sample of 10 callers and calculates their mean wait time x bar. Is it possible to calculate the SD of x bar. If it is, do it. If it isn't explain why.

Yes, it seems reasonable to assume that the sample of 10 is less than 10% of the entire population calls. Ox = 5 / root of 10

Companies are interested in the demographics of those who listen to the radio programs they sponsor. A radio station had determined that only 20% of the listeners phoning in to a morning talk show are male. The station management wonders if adding a male host to the program will increase the proportion of callers who are male. After adding the male host, the station records the gender of 200 people who phone into the program during a particular week. The station is willing to view these 200 callers as an SRS from the population of all those who call in to the program. In fact, during this particular week, 50/200 callers were male. Does this provide sufficient evidence to suggest that the proportion of male callers has increased from 20%?

The probability of getting 50+ males in 200 callers if the true proportion of males is still .20 is (.25, 99999999, .20, .028) = .0367; roughly one out of 25 times, we will get this many or more male callers. This is probably unusual enough to suggest that the true proportion of male listeners if higher than .20

What is there an important connection between?

The sample proportion and the number of successes in the sample

What must we be able to describe in order to perform statistical inference?

The sampling distribution of possible statistic values

What does the shape of the distribution of x bar depend on?

The shape of the population distribution

What does the variability of a statistic in repeated sampling not depend on?

The size of the population

What is the spread determined by?

The size of the random sample

What is the variability of a statistic described by?

The spread of its sampling distribution

What is sampling variability?

The value of a statistic varies in repeated random sampling

Why do larger samples have an advantage over smaller?

They are more likely to produce an estimate close to the true values of the prameter

What does the central limit theorem allow us to do?

Use normal probability calculations to answer questions about sample means from many observations even when the population distribution is not normal

What does hw process of statistical inference involve?

Using informatiom from a sample to draw conclusions about a wider population

Companies are interested in the demographics of those who listen to the radio programs they sponsor. A radio station had determined that only 20% of the listeners phoning in to a morning talk show are male. The station management wonders if adding a male host to the program will increase the proportion of callers who are male. After adding the male host, the station records the gender of 200 people who phone into the program during a particular week. The station is willing to view these 200 callers as an SRS from the population of all those who call in to the program. What assumption are you making when you use the formula for the SD of p hat in this setting?

We can use this formula if we assume that there are more than 10 (200) = 2000 listeners who call in to the program

Teenagers send many messages - recent polls cite medians of more than 50 per day. Consider a large population of teenagers for whom the distribution of the number of texts sent per day is strongly skewed to the right. The min is 0, q1 is 20, median is 55, q3 is 140 and max is 250. Suppose we take random samples of size 32 from this population and calculate q1 for each of the samples. Is the sample q1 an unbiased estimator of the population q1?

We know from the five number summary that the true population for q1 is 20. The mean of the stimulated sampling distribution appears to be higher than 20 and therefore the sample first quartile is a biased estimator

To make sense of sampling variability, what can we ask?

What would happen if we took many samples?

What is bias?

When our aim is off and we consistently miss in the same direction

What is the formula for p hat?

count of successes / size of sample

what is the formula for SD?

square root of np(1-p)

What is the equation for phat SD?

square root of p(1-p) / n

Companies are interested in the demographics of those who listen to the radio programs they sponsor. A radio station had determined that only 20% of the listeners phoning in to a morning talk show are male. The station management wonders if adding a male host to the program will increase the proportion of callers who are male. After adding the male host, the station records the gender of 200 people who phone into the program during a particular week. The station is willing to view these 200 callers as an SRS from the population of all those who call in to the program. For the moment, assume that the addition of the male host has no effect on the proportion of caller who are male. If is the proportion of callers in the sample who are male, what are the mean and SD of the sampling distribution of p hat?

mean = .2 SD = square root of (.2)(.8) / 200 = /.028

What is the symbol for population mean?

mu

What is the formula for the 10% condition?

n < 1/10 N

A friend has offered to play a game w you that involves flipping a coin that he has provided. Since a flip of heads will be to his advantage, you want to test the coin for fairness before you begin to play. Your friend is willing to let you flip the coin 50 times to determine if the probability of getting heads is actually .50, as it should be if the coin in fair. Assume for the moment that the coin is far. You flip the coin 5 times and get 30 heads. Do you risk insulting your find by refusing to play with this coin? Explain.

normal cdf = (.6, 9999999, .5, .071) = .0793; Roughly 2 out of 2t times, we will get this many or more heads. This is not unusual enough to risk your friend's good will by accusing him of an unfair coin

What is the formula for mean?

np

What are the two formulas for normal condition?

np > 10 n(1-p) > 10

If x is the mean of a SRS of size n drawn from a large population with mean u and SD o, what is the SD of the distribution?

o / square root of n

What is the mean of the distribution?

p

What is the symbol for population proportion?

p

When choosing a SRS of size n and a population of size N with proportion p of successes with p hat being the sample proportion of successes, what is the mean of p hat?

p

What is used to estimate the unknown parameter p?

p hat