Elementary Statistics
Chapter 10.1 Learning the Language of Hypothesis Testing
Determine the null and alternative hypothesis
State conclusions to hypothesis tests
A hypothesis is a statement regarding a characteristic of one or more populations
In this context, we will be looking at a hypothesis regarding a single population parameter
Examples of Claims regarding a characteristic of a Single Population
In 2008, 62% of American adults regularly volunteered their time for charity work. A researcher believes that this percentage is different today.
Hypothesis testing is a procedure, based on sample evidence and probability, used to test statements regarding a characteristic of one or more populations.
Steps in Hypothesis Testing
Make a statement regarding the nature of the population
Collect evidence (sample data) to test the probability
Analyze the data to assess the plausibility of the statement
The null hypothesis, denoted H0, Is a statement to be tested. The null hypothesis is a statement of no change, no effect, or no difference and is assumed true until evidence indicates otherwise
The alternative hypothesis, denoted H1, is a statement that we are trying to find evidence to support
In this chapter there are three ways to set up the null and alternative hypotheses
Equal versus not equal hypothesis (two-tailed test)
H0: parameter= some value
H1: parameter doesn’t equal some value
Equal versus less than (left tailed test)
H0: parameter = some value
H1: parameter < some value
Equal versus greater than (right tailed)
H0 parameter = some value
H1: parameter > some value
In other words the null hypothesis is a statement of status quo or no difference and always contains a statement of equality. The null hypothesis is assumed to be true until we have evidence to the contrary. We seek evidence that supports the statement in the alternative hypothesis.
We never “accept” the null hypothesis, because without having access to the entire population we don’t know the exact value of the parameter stated in the null. Rather, we j say that we do not reject the null hypothesis. This is just like the court system. We never declare a defendant “innocent” but rather say the defendant is not guilty.
10.2 Hypothesis Test for a Population Proportion
Test the hypotheses about a population proportion
Test hypotheses about a population proportion using the binomial probability distribution.
Recall
The best point estimate of p, the proportion of the population with a certain characteristic is given by p hat = x/n
where x is the number of individuals in the sample with the specified characteristic and n is the sample size.
The sampling distribution of p hat is approximately normal with mean =p and standard deviation
Provided that the following requirements are satisfied:
The sample is simple random sample
np(1-p) > or equal to 10
The sample values are independent of each other
Testing Hypotheses Regarding a Population Proportion, p.
To test hypotheses Regarding a Population Proportion, we can use the steps that follow, provided that:
The sample is obtained by simple random sampling
np0(1-p0) > or equal to 10
The sampled values are independent of each other.
If the P-value < a, reject the null hypothesis
testing a Hypothesis about a population proportion Large Sample Size
In 1997, 46% of Americans said they did trust the media “when it comes to reporting the new fully accurately and fairly”. In a 2007 poll of 1010 adults nationwide, 525 stated they did not trust the media. At the a=0.05 significance level, is there evidence to support claim that the percentage of American do not trust the media to report fully accurately has increased since 1997
Solution
We want to know if p>0.46. First, we must verify the requirements to preform hypothesis test
This is a simple random sample
np0(1-p0) = 1010(0.46)(1-0.46)= 250.9> 10
Since the sample size is less than 5% of the population size, the assumption of independence is met.
10.3 Hypothesis Tests for a Population Mean
Test hypotheses about a mean
Understand the difference between statistical significance and practical significance
To test hypotheses regarding the population mean assuming the population standard deviation is unknown, we use the t-distribution rather than the Z-distribution..
Properties of the t-distribution
The t-distribution is different for the degrees of freedom
The t-distribution is centered at 0 and is symmetric about 0.
The area under the curve is 1. Because of the symmetry, the area under the curve to the right of 0 equals thee curve to the of of 0 equal 1/2.
As t increases (or decreases) without bound, the graph approaches, but never equals 0.
The area in the tails of the t-distribution is a little greater than the area in the tails of the standard normal distribution because using s as an estimate of o introduces more variability to the t-statistic
As the sample size n increases, the density curve of t gets closer to the standard normal density curve. This result occurs because as the sample size increases, the values of s gets closer to the values of o by the law of large numbers
Testing Hypotheses Regarding a Populattion Mean
To test hypotheses regarding the population mean, we use the following steps, provided that
The sample is obtained using simple random sampling
The sample has no outliers, and the population from which the sample is drawn normally distributed or the sample size is large n>30
The sampled values are independent of each other
The procedure is robust, which means that minor departures from normality will not adversely affect the results of the test. However, for small samples, if the data have outliers, the procedure shouldn’t be used
10.4 Hypothesis test for a Population Standard deviation
Test hypotheses about a population standard deviation
Chi-square Distribution
If a simple random sample of size n is obtained from a normally distributed population with a mean and standard deviation, then use a ci square distribution
Characteristics of the Chi-Square Distribution
It is not symmetric
The shape of the chi-square distribution depends on the degrees of freedom
As the number of degrees of freedom increases, the chi-square distribution becomes more nearly symmetric.
The values of X2 are nonnegative (greater than or equal to 0)
Testing Hypotheses about a Population Variance or Standard Deviation
The sample is obtained using simple random sampling or from a randomized experiment
The population is normally distributed
The procedures in 10.4 aren’t robust.
This means if the data analysis indicates sthat the variable does not come from a popukation that is normally distributed, the procedure presented in this section is not valid.
Chapter 10.1 Learning the Language of Hypothesis Testing
Determine the null and alternative hypothesis
State conclusions to hypothesis tests
A hypothesis is a statement regarding a characteristic of one or more populations
In this context, we will be looking at a hypothesis regarding a single population parameter
Examples of Claims regarding a characteristic of a Single Population
In 2008, 62% of American adults regularly volunteered their time for charity work. A researcher believes that this percentage is different today.
Hypothesis testing is a procedure, based on sample evidence and probability, used to test statements regarding a characteristic of one or more populations.
Steps in Hypothesis Testing
Make a statement regarding the nature of the population
Collect evidence (sample data) to test the probability
Analyze the data to assess the plausibility of the statement
The null hypothesis, denoted H0, Is a statement to be tested. The null hypothesis is a statement of no change, no effect, or no difference and is assumed true until evidence indicates otherwise
The alternative hypothesis, denoted H1, is a statement that we are trying to find evidence to support
In this chapter there are three ways to set up the null and alternative hypotheses
Equal versus not equal hypothesis (two-tailed test)
H0: parameter= some value
H1: parameter doesn’t equal some value
Equal versus less than (left tailed test)
H0: parameter = some value
H1: parameter < some value
Equal versus greater than (right tailed)
H0 parameter = some value
H1: parameter > some value
In other words the null hypothesis is a statement of status quo or no difference and always contains a statement of equality. The null hypothesis is assumed to be true until we have evidence to the contrary. We seek evidence that supports the statement in the alternative hypothesis.
We never “accept” the null hypothesis, because without having access to the entire population we don’t know the exact value of the parameter stated in the null. Rather, we j say that we do not reject the null hypothesis. This is just like the court system. We never declare a defendant “innocent” but rather say the defendant is not guilty.
10.2 Hypothesis Test for a Population Proportion
Test the hypotheses about a population proportion
Test hypotheses about a population proportion using the binomial probability distribution.
Recall
The best point estimate of p, the proportion of the population with a certain characteristic is given by p hat = x/n
where x is the number of individuals in the sample with the specified characteristic and n is the sample size.
The sampling distribution of p hat is approximately normal with mean =p and standard deviation
Provided that the following requirements are satisfied:
The sample is simple random sample
np(1-p) > or equal to 10
The sample values are independent of each other
Testing Hypotheses Regarding a Population Proportion, p.
To test hypotheses Regarding a Population Proportion, we can use the steps that follow, provided that:
The sample is obtained by simple random sampling
np0(1-p0) > or equal to 10
The sampled values are independent of each other.
If the P-value < a, reject the null hypothesis
testing a Hypothesis about a population proportion Large Sample Size
In 1997, 46% of Americans said they did trust the media “when it comes to reporting the new fully accurately and fairly”. In a 2007 poll of 1010 adults nationwide, 525 stated they did not trust the media. At the a=0.05 significance level, is there evidence to support claim that the percentage of American do not trust the media to report fully accurately has increased since 1997
Solution
We want to know if p>0.46. First, we must verify the requirements to preform hypothesis test
This is a simple random sample
np0(1-p0) = 1010(0.46)(1-0.46)= 250.9> 10
Since the sample size is less than 5% of the population size, the assumption of independence is met.
10.3 Hypothesis Tests for a Population Mean
Test hypotheses about a mean
Understand the difference between statistical significance and practical significance
To test hypotheses regarding the population mean assuming the population standard deviation is unknown, we use the t-distribution rather than the Z-distribution..
Properties of the t-distribution
The t-distribution is different for the degrees of freedom
The t-distribution is centered at 0 and is symmetric about 0.
The area under the curve is 1. Because of the symmetry, the area under the curve to the right of 0 equals thee curve to the of of 0 equal 1/2.
As t increases (or decreases) without bound, the graph approaches, but never equals 0.
The area in the tails of the t-distribution is a little greater than the area in the tails of the standard normal distribution because using s as an estimate of o introduces more variability to the t-statistic
As the sample size n increases, the density curve of t gets closer to the standard normal density curve. This result occurs because as the sample size increases, the values of s gets closer to the values of o by the law of large numbers
Testing Hypotheses Regarding a Populattion Mean
To test hypotheses regarding the population mean, we use the following steps, provided that
The sample is obtained using simple random sampling
The sample has no outliers, and the population from which the sample is drawn normally distributed or the sample size is large n>30
The sampled values are independent of each other
The procedure is robust, which means that minor departures from normality will not adversely affect the results of the test. However, for small samples, if the data have outliers, the procedure shouldn’t be used
10.4 Hypothesis test for a Population Standard deviation
Test hypotheses about a population standard deviation
Chi-square Distribution
If a simple random sample of size n is obtained from a normally distributed population with a mean and standard deviation, then use a ci square distribution
Characteristics of the Chi-Square Distribution
It is not symmetric
The shape of the chi-square distribution depends on the degrees of freedom
As the number of degrees of freedom increases, the chi-square distribution becomes more nearly symmetric.
The values of X2 are nonnegative (greater than or equal to 0)
Testing Hypotheses about a Population Variance or Standard Deviation
The sample is obtained using simple random sampling or from a randomized experiment
The population is normally distributed
The procedures in 10.4 aren’t robust.
This means if the data analysis indicates sthat the variable does not come from a popukation that is normally distributed, the procedure presented in this section is not valid.
