Statistics Exam 2 (Norwood)

At a cat race, the average speed of a racing cat is known to be normally distributed with a standard deviation of 0.5. A sample of 20 cats is taken and their average speed is found to be 10.20. Develop a 90% confidence interval for the population mean cat speed.

(10.194, 10.205)

Find the confidence interval when sample mean is 15, n=35, SD=2.3, at a 95% confidence level.

(14.981, 15.019)

A Goofy Professor decided to test the hypothesis that the average time a students are spending on homework is more than 33 minutes. A random sample of 50 students were selected and they averaged 37 minutes. It is believed that the population standard deviation is 4 minutes. The α is set to 0.05. The p-value for this hypothesis test would be closest to __________. (Hint you are trying to find P(x>37) you will need to use the Z table to look up the correct probability and use the correct procedure when you have the > sign.)

.0011

If a random variable is normally distributed what is the probability the normal random variable is less than its mean?

0.5

With Chapter 6 we learned continuous random variables and we were able to find the probability greater than a point by finding the probability and doing what?

1- z score of point to percentage = probability greater than a point

Z score of 90% confidence interval

1.645

Z score of 95% confidence interval

1.96

Z score of 99% confidence interval

2.576

From chapter 6, when you go one SD from the mean what % of the sample do you capture? What about when you go 2 SD from the mean?

68%, 95%

Confidence Interval excel codes (for 95% interval)

=CONFIDENCE.T(0.05, standard deviation, mean)

As the sample size goes up, what happens to the margin of error?

As sample size increases, the margin of error decreases

What happen as the sample size increases to the standard errors?

As the sample size becomes larger, the standard error will become smaller

What happens to the distribution and results as your sample size gets larger?

As the sample sizes increase, the variability of each sampling distribution decreases

The probability the variable assumes a value with an interval, P(a≤X≤b) is defined as.....

Cumulative distribution function (cdf) A function giving the probability that a random variable is less than or equal to a specified value.

The mean or the expected value of X for the continuous uniform distribution is computed as

E(X)=μ=(a+b)/2, where a and b are lower and upper limits of values..

What is the relationship between the expected value of the sample mean and the expected value of the population?

Expected value of sample mean = Population mean

Know how to set up the null and alternative hypothesis.

Ho = Ha (≠ or < or >)

As the sample size increases what happens to the width of the confidence interval? What happens to the standard errors?

Increasing the sample size decreases the width of confidence intervals, because it decreases the standard error.

Know the difference between a one and two tailed test.

One-tailed tests allow for the possibility of an effect in one direction. Two-tailed tests test for the possibility of an effect in two directions—positive and negative.

What is stratified random sampling?

Population is split into CATEGORIES and then pure random selection of certain number of people within each category

Regarding to qualitative data and the population/sample proportion what conditions must be met?

Random: The data needs to come from a random sample or randomized experiment. Normal: the sampling distribution of p-hat (probability of sample proportion) needs to be approximately normal — needs at least 10 expected successes and 10 expected failures Independent: Individual observations need to be independent. If sampling without replacement, our sample size shouldn't be more than 10% percent of the population.

The standard deviation of X is computed as

SD(x)= SQRT((b-a)^2)/(12))

If you do not have a set upper and lower limit for statistical quality control what is the general rule/formula to find them?

Subtract the upper class limit for the first class from the lower class limit for the second class divided by 2 and subtract/add for lower and upper of each class

What are similarities and differences between the T-distribution and the Z-distribution?

T distribution is sensitive to size and used for smaller samples (less than 30) both have a mean of 0 and sd of 1 Z test view the average of data

In regards to confidence intervals you use the T table when _____ is unknown and the Z table when the _______ is known. (Think about the confidence interval case study)

T table: when sample size is less than 30/population standard deviation Z table: population's standard deviation and mean

Type I and Type II Errors

Type 1: When you reject the null, but the null is true Type 2: occurs when you accept the null, but the null is true

margin of error formula

Z score or T score * standard deviation

What is the empirical rule from Ch. 7?

a normal distribution (bell shape curve) will always fall within 3 standard deviations 1 (68%) 2 (95%) 3 (99.7%)

Be able to manually calculate a confidence interval if you are given the sample mean, confidence level, sample size, and SD.

confidence interval = sample mean ± confidence level value (sample standard deviation/sqrt(n))

If given a Z score can you find the probability on the Z-Table?

example: Z score = 1.09 left column = 1.0 top column = 0.09

If an exponential distribution has the rate parameter λ = 4, then its expected value found how

expected value is 1/lambda expected value is 1/4th

What are some examples of a continuous random variable?

height, weight, distance

With a 2 tailed test what do you have to do to find the probability?

higher and lower in Ha

What is a continuous random variable?

infinitely many values; infinite number of uncountable outcomes

As the confidence interval increases from 90% to 95% to 99% what happens to the width of the confidence interval? What happens to the standard errors?

it gets wider, standard deviations get smaller

As the population standard deviation goes up, what happens to the width of the confidence interval?

it increases

If you have a positive Z score and it is a right tailed test what do you have to do when you look up the probability on the Z table?

it is above the mean average and is x standard deviations above the mean

Remember with probability < and > rules on finding the correct probability.

less than (<) = z score to percentage greater than (>) = 1 - z score to percentage

What is the probability that any random number is more/less than the mean?

less than = decimals below mean greater than = decimals above mean less than or equal to = decimals at and below mean greater than or equal to = decimals at and above mean

If given 2 data end points 40 and 300 find the expected mean and SD.

mean = 170 standard deviation = 75.06

What is a z-score?

number of standard deviations away from the mean

If given numbers, can you find the sample proportion? (Hint Ch. 7 finding if it is not given)

p hat = x / n x = number of successes n = sample size example: a sample of 200 people entering the store, 78 make a purchase, p hat = 78/200 = 0.39

confidence interval formula

point estimate +/- margin of error(critical value x standard error)

μ

population mean (mu)

σ

population standard deviation (sigma)

What is a confidence interval?

range of values in which a specified probability of the means of repeated samples would be expected to fall

x̄ (x-bar)

sample mean

n

sample size

Example of sample bias

survey of high school students to measure teenage use of illegal drugs will be a biased sample because it does not include home-schooled students or dropouts

If given numbers can you find the P-Value for a Hypothesis test?

take Ho/Ha < -1 > +1 ≠ double value found

What is the central limit theorem?

the sum or mean of a large number of independent observations from the same underlying distribution has an approximate normal distribution

What is sample bias

the tendency of a sample statistic to systematically overestimate or underestimate a population parameter

When the test is right tailed you have to do what differently compared to a left tailed test. o When P(x<2.5) compared to the difference when you have P(x>2.5)

when HA contains: < = left tailed > = right tailed

When a soda machine is properly calibrated, it takes an average of 6 seconds to send out a soda once purchased, with a standard deviation of 2 seconds. For a simple random sample of n = 36 units, the sample mean is found to be 8 seconds per unit. When the machine is properly calibrated, what is the Z-Score from a sample this size for x bar = 8 seconds?

x bar - population mean (divided by) standard deviation/square root(n) 8 - 6 (divided by) 2/square root (36) 2 (divided by) 0.333 z score = 6

What is the z-score formula?

z = (x - μ)/σ z = (x-mean)/standard deviation