Statistics Exam #3

8.1 A simple random sample of size n= 40 is obtained from a population with μ= 50 and σ= 4. Does the population need to be normally distributed for the sampling distribution of x̅ to be normally distributed? Why? What is the sampling distribution of x̅?

No because n> 30; the sampling distribution of x̅ is normally distributed.

What does area equal?

Probability

What is p̂?

Sample percentage

What is the central limit theorem?

States that x̅ is a normal distributed random variable is: 1. n> 30 2. x is normally distributed

9.2 As the number of degrees of freedom in the t-distribution increases, the spread of the distribution _____________.

decreases

What tend to give rise to normal distributions?

Summing and averaging processes; also manufacturing and biological processes

8.2 The _________ ___________, denoted p̂, is given by the formula p̂= _________, where x is the number f individuals with a specified characteristic in a sample of n individuals.

sample percentage x/n

8.1 The _________ ________ of the sample mean, x̅, is the probability distribution of all possible values of the random variable x̅ computed from a small size of n from a population with mean μ and standard deviation σ.

sampling distribution

8.1 The standard deviation of the sampling distribution of x̅, σx̅, is called the ____________ ___________ of the ___________.

standard error, mean

7.2 A random variable z that is normally distributed with mean μ= 0 and standard deviation σ= 1 is said to have the ___________ ___________ ____________.

standard normal distribution

8.1 True or False: The distribution of the sample mean,x̅, will be normally distributed if the sample is obtained from a population that is normally distributed, regardless of the sample size.

true; if random variable x is normally distributed, the sampling distribution of the sample mean x̅ is normally distributed.

8.1 Suppose a sample size of n is drawn from a large population with mean μ and standard deviation σ. The sampling distribution of x̅ has mean μx̅= _____________ and standard deviation σx̅= _______________.

x σ/ √n

How do you find p̂?

x/ n

How do we find a normal probability plot on the calculator?

-Enter data into L1 (stat, edit) -Turn on stat plot (2nd, y=) -Use the bottom right chart option (L1, x-axis) -Zoom, zoom stat (#9)

What z-score equals 90% confidence?

1.645

9.1 As the sample size used to obtain a confidence interval increases, the margin of error ____________.

decreases

How can we tell if something is a normal distribution when looking at a normal probability plot?

The points on the plot will look linear.

How can we tell if something is not a normal distribution when looking at a normal probability plot?

The points on the plot will make a curve

9.1 As the level of confidence of a confidence interval increases, the margin of error ____________.

increases

What does n and N stand for?

n= sample size N= population size

7.3 A _____________ ___________ ________ is a graph that plots observed date versus normal scores.

normal probability plot

7.1 True or False: The normal curve is symmetric about its mean, μ

true

8.2 True or False: The mean of the sampling distribution of p̂ is p.

true

What is the z-score equation?

z= (x-μ)/ σ

What is the mean of x̅ (μx̅) and the standard deviation of x̅ (σx̅)?

μx̅= μ σx̅= σ/ √n

7.2 If X is a normal random variable with mean 40 and standard deviation 10 and P (X>45) = 0.3085, then P (X< 35)= ____________.

0.3085

8.2 In a town of 500 households, 220 have a dog. The population proportion of dog owners in this town (expressed as a decimal) is p= __________.

0.44 (220/500)

What are the two interpretations of area?

1. Area to the right of __________ equals the probability that a randomly selected ________ is larger than _______. 2. Area to the right of _________ is the percentage of _________ larger than ____________.

What are the properties of a normal distribution?

1. Bell- shaped curve (symmetric) 2. Mean= median= mode (highest point on curve) 3. Total area under curve = 1

What z-score equals 99% confidence?

2.576

What z-score equals 95% confidence?

1.96

7.1 The area under the normal curve to the right of μ equals _________.

1/2

What is the symbol for margin of error?

E

How can we tell if p̂ is normal?

For a binomial experiment with n trials and probability of success p, both 1. n≤ 0.05N 2. np(1-p) ≥ 10 then the random variable p̂ is approximately normally distributed.

What is a standard normal distribution?

It is the you have a mean of 0 and a standard deviation of 1 (μ= 0 and σ= 1)

What side of the curve does the standard normal distribute chart give us the number to?

The left side.

8.2 Describe the circumstances under which the shape of the sampling distribution of p̂ is approximately normal.

The sampling distribution of p̂ is approximately normal when n ≤ 0.005 N and np(1-p)≥ 10

How do you write an interpretation for the confidence interval of population mean?

We are _____ % confident that population mean of ______________ is between ________(unit) and _________(unit).

How do you write an interpretation for the confidence interval for population/ proportion p?

We are _____ % confident that the percentage of ______________ is between ________ %and _________%.

What does less than mean?

We shade to the left of side.

What does greater than mean?

We shade to the right side.

8.2 True or False: The population proportion and sample proportion always have the same value

false

8.1 True or False: The distribution of the sample mean, x̅, will be normally distributed if the sample is obtained from a population that is not normally distributed, regardless of the sample size.

false; sample size would have to be greater than 30

9.2 True or False: To construct a confidence interval about the mean, the population from which the sample is drawn must be approximately normal.

false; the population does not need to be normally distributed if the sample size is sufficiently large

How do you express you answer when solving a confidence interval equation for proportion p?

negative margin of error < p < positive margin of error

7.1 The area under a normal curve can be interpreted as __________ or ____________.

percentages or probabilities

9.1 A __________ _________ is the value of a statistic that estimates the value of a parameter

point estimate ** in confidence interval for population it is p̂, for confidence interval for μ it is x̅

7.1 A ___________ ____________ __________ is an equation used to compute probabilities of continuous random variables.

probability density function

7.2 The notation zα is the z-score such that the area under the standard normal curve to the right of zα is ____________.

α

9.2 The notation tα is the t-value such that the area under the t-distribution to the right of tα is ____________.

α

What is the mean of p̂ (μp̂) and the standard deviation of p̂ (σp̂)?

μp̂= P (population) σp̂=√p(1-p)/n