Statistics Chapter 7 Flashcards

Questions and Answers

What is a number that describes some characteristic of a sample?

statistic

What is a number that describes some characteristic of the population?

parameter

X̄ is the and estimates the .

sample mean, population mean

P̂ is the and estimates the .

sample proportion, population proportion Signup and view all the answers

Sx is the and estimates the .

sample SD, population SD Signup and view all the answers

For sample proportions, when n is small and P̂ is close to 0, what is the shape of the sampling distribution of P̂?

right skewed Signup and view all the answers

For sample proportions, when n is small and P̂ is close to 1, what is the shape of the sampling distribution of P̂?

left skewed Signup and view all the answers

When n is large and/or P̂ is close to 0.5, what is the shape of the sampling distribution of P̂?

approximately Normal Signup and view all the answers

What must happen in order for the SD of the sampling distribution to be √p(1-p)/n?

the 10% condition must be met Signup and view all the answers

The sampling distribution of P̂ is approximately Normal as long as what condition is true?

the Large Counts Condition Signup and view all the answers

What must happen for the SD of the sampling distribution of X̄ to be σ/√n?

the 10% condition must be met Signup and view all the answers

What does the value of the standard deviation of X̄ measure?

the typical distance between the sample mean X̄ and the population mean Signup and view all the answers

If the population distribution P̂ is approximately normal, what is the shape of the sample distribution?

approximately Normal Signup and view all the answers

For a sample distribution, when n is large, the sample mean X̄ is?

approximately normal Signup and view all the answers

In order to use the formula σ/√n to calculate the standard deviation of the sampling distribution of the sample mean, which of the following conditions must be met?

iii only Signup and view all the answers

The central limit theorem refers to which of the following characteristics of the sampling distribution of the sample mean?

Regardless of the shape of the population distribution, the sampling distribution of the sample mean from sufficiently large samples will be approximately Normally distributed. Signup and view all the answers

A statistic is said to be unbiased if:

The mean of its sampling distribution is equal to the true value of the parameter being estimated. Signup and view all the answers

The mean weight of olives classified as 'colossal' is 7.7 grams. If the crop is approximately normally distributed, which of the following probabilities represents the likelihood that the mean weight of a random sample of 3 olives is greater than 8 grams?

P(z > 0.5 - 0.46 / √((0.46)(0.54) / 100)) Signup and view all the answers

The distribution of bulb burnout times is strongly skewed to the right. According to the central limit theorem, what does this imply?

The mean burnout time for a large number of bulbs has a distribution that is close to Normal. Signup and view all the answers

The chipmunk population in a certain area has a mean weight of 84 gm and a SD of 18 gm. A wildlife biologist weighs 9 chipmunks. Which of the following best describes the sampling distribution of means?

μx̅ = 84; σx̅ = 6; shape of distribution unknown Signup and view all the answers

According to the Dupont 2012 global automotive report, 25% of all cars manufactured in 2012 were white. In a sample of 100 cars parked, 19% were white. Which statement is true?

23% is a parameter, 19% is a statistic. Signup and view all the answers

What characteristics should the best statistic for estimating a parameter have?

low bias, low variability Signup and view all the answers

In a large population of adults, the mean IQ is 112 with a standard deviation of 20. What is the sampling distribution of the sample mean IQ for 20 randomly selected adults?

approximately Normal, mean 112, SD 1.414 Signup and view all the answers

In a study of acid rain effects, 100 trees were examined, and 40% showed damage. Which statement is correct?

The sampling distribution of the sample proportion of damaged trees is approximately Normal. Signup and view all the answers

Which of the following statements is/are true when taking a SRS from a large population?

i, ii, and iii Signup and view all the answers

In order to use the formula √p(1-p)/n, which conditions must be met?

i only Signup and view all the answers

Suppose you take a random sample of size 25 from a population with mean of 130 and a standard deviation of 20. Your sample has a mean of 128 and a standard deviation of 19.8. Which has a mean of 130 and a standard deviation of 4?

The sampling distribution of the sample mean. Signup and view all the answers

Study Notes

Statistics Concepts

A statistic is a number describing a characteristic of a sample.
A parameter is a number that describes a characteristic of a population.
X̄ represents the sample mean, estimating the population mean.
P̂ denotes the sample proportion, estimating the population proportion.
Sx symbolizes the sample standard deviation, estimating the population standard deviation.

Sampling Distributions

When sample proportions (P̂) are calculated, if the sample size ( n ) is small and P̂ is close to 0, the distribution is right skewed.
Conversely, if P̂ is close to 1 in a small sample, the distribution becomes left skewed.
For large ( n ) or when P̂ is around 0.5, the sampling distribution of P̂ is considered approximately Normal.
The 10% condition must be met for the standard deviation of the sampling distribution to be calculated as ( \sqrt{p(1-p)/n} ).
The Large Counts Condition establishes that the sampling distribution of P̂ is approximately Normal if certain thresholds are met.

Sample Mean Distribution

The standard deviation of the sample mean (X̄) will be ( \sigma/\sqrt{n} ) if the 10% condition is satisfied.
This standard deviation (σx̄) measures the typical distance between the sample mean and the population mean.
If the population distribution of P̂ is approximately normal, the sample distribution will also be approximately Normal.
For large ( n ), the sample mean X̄ is approximately normal.

Central Limit Theorem

The Central Limit Theorem asserts that for sufficiently large samples, the sampling distribution of the sample mean will be approximately Normally distributed, irrespective of the population distribution shape.
The unbiased statistic has a mean of its sampling distribution equal to the true value of the parameter being estimated.

Light Bulb Burnout Times

For a distribution of burnout times that is strongly right skewed, with larger sample sizes, the mean burnout time's distribution tends towards Normal.

Chipmunk Weight Example

For a wildlife biologist's sample of 9 chipmunks weighing on average 84 gm (mean) with an SD of 18 gm, the sampling distribution mean is 84 gm, but the SD is unknown, indicating it is approximately Normal.

Estimation and Sampling

In a car sample where 19% of cars are white from a population where 25% are white, the adequate characterization is knowing 19% is a statistic and 25% is a parameter.
An effective statistic for estimating a parameter should have low bias and low variability.
Given a population mean IQ of 112 with a standard deviation of 20, the correct approximation for a sample mean from 20 adults results in an SD of approximately 1.414.

Acid Rain Study

In examining the impact of acid rain on trees, the sample proportion indicating damage is considered approximately Normal.

Sampling Conditions

For any sample, the requirement that np ≥ 10 and n(1-p) ≥ 10 together form conditions for approximating sampling distributions.
The sampling distribution of the sample mean from a sample size of 25 will have the same mean (130) with a reduced standard deviation (4), signifying the link to its population characteristics.

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Test your knowledge of key concepts from Statistics Chapter 7 with these flashcards. Each card contains a term along with its definition, helping you grasp important statistical vocabulary. Perfect for students looking to reinforce their understanding of statistics.