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Transcript
Statistics 400 - Lecture 9
 Today: Sections 8.3
 Read 8.3 and 8.4 for next day
 VERY IMPORTANT SECTIONS!!!
Confidence Intervals for the Mean
 Last day, introduced a point estimator…a statistic that estimates a
population parameter
 Often more desirable to present a plausible range for the
parameter, based on the data
 We will call this a confidence interval
 Ideally, the interval contains the true parameter value
 In practice, not possible to guarantee because of sample to sample
variation
 Instead, we compute the interval so that before sampling, the
interval will contain the true value with high probability
 This high probability is called the confidence level of the interval
Confidence Interval for  for a Normal
Population
 Situation:
 Have a random sample of size n from
N (  , )
 Suppose value of the standard deviation is known
 Value of population mean is unknown
 Last day we saw that 100(1   )% of sample means will fall in the
interval:

 



z
,


z
 /2
 /2

n
n 
 Therefore, before sampling the probability of getting a sample
mean in this interval is (1   )
 Equivalently,

 

P   z / 2
 X    z / 2   (1   )
n
n

 Equivalently,

 

P X  z / 2
   X  z / 2   (1   )
n
n

 The interval below is called a 100(1   )% confidence interval for

 

X

z
,
X

z
 /2
 /2

n
n 

Example
 To assess the accuracy of a laboratory scale, a standard weight
known to be 10 grams is weighed 5 times
 The reading are normally distributed with unknown mean and a
standard deviation of 0.0002 grams
 Mean result is 10.0023 grams
 Find a 90% confidence interval for the mean
Interpretation
 What exactly is the confidence interval telling us?
 Consider the interval in the previous example. What is the
probability that the population mean is in that particular interval?
 Consider the interval in the previous example. What is the
probability that the sample mean is in that particular interval?
Large Sample Confidence Interval for
 Situation:
 Have a random sample of size n (large)
 Suppose value of the standard deviation is known
 Value of population mean is unknown

 If n is large, distribution of sample mean is
 Can use this result to get an approximate confidence interval for the
population mean
 When n is large, an approximate
for the mean is:
100(1   )% confidence interval
Large Sample Confidence Interval for 
(unknown standard deviation)
 Situation:
 Have a random sample of size n (large)
 Suppose value of the standard deviation is unknown
 Value of population mean is unknown
 When n is large, replacing the population standard deviation with
the sample estimate gives a good approximation
 When n is large and the population standard deviation is unknown,
an approximate 100(1   )% confidence interval for the mean is:
Example (8.19)
 Amount of fat was measured for a random sample of 35
hamburgers of a particular restaurant chain
 Sample mean and sample standard deviation were found to be 30.2
and 3.8 grams
 Find a 95% confidence interval for the mean fat content of
hamburgers for this chain
Changing the Length of a Confidence
Interval
 Can shorten the length of a confidence interval by:
 Using a difference confidence level
 Increasing the sample size
 Reducing population standard deviation
Hypothesis Testing
 Hypothesis testing is a statistical technique to test if a conjecture
about a population parameter is true
 Has 4 Main Steps:
 Null and Alternate Hypotheses
 Test Statistic
 P-Value
 Decision based on pre-specified error rate
Example
 Heights of one-year-old girls normally distributed with mean 30
inches and standard deviation of 1.2 inches
 Company claims taking 500 mg of Vitamin C makes the girls taller
1. Hypotheses
 Begin by making an assumption of no change
 This statement is called the null hypothesis (H0)
 Test will be designed to assess evidence against H0
 Hypothesis we suspect is true is called alternate hypothesis (H1)
 Assume H0 is true, collect data and see if there is evidence against
H0 and in favor of H1
Example
 Heights of one-year-old girls normally distributed with mean 30
inches and standard deviation of 1.2 inches
 Company claims taking 500 mg of Vitamin C makes the girls taller
 H0:
 H1: