Chapter 9 Hypothesis Testing.

Chapter 9 Hypothesis Testing

Hypothesis Testing 9.1 Null and Alternative Hypotheses and Errors in Testing 9.2 z Tests about a Population Mean s Known 9.3 z Tests about a Population Mean s Known 9.4 t Tests about a Population Proportion

Hypothesis Testing Continued
9.5 Type II Error Probabilities and Sample Size Determination (Optional) 9.6 The Chi-Square Distribution (Optional) 9.7 Statistical Inference for a Population Variance (Optional)

Null and Alternative Hypotheses and Errors in Hypothesis Testing
The null hypothesis, denoted H0, is a statement of the basic proposition being tested. The statement generally represents the status quo and is not rejected unless there is convincing sample evidence that it is false. The alternative or research hypothesis, denoted Ha, is an alternative (to the null hypothesis) statement that will be accepted only if there is convincing sample evidence that it is true.

Example 9.1: Trash Bag Case
Tests show the trash bag has a mean breaking strength µ close to but not exceeding 50 lbs The null hypothesis H0 is that the new bag has a mean breaking strength that is 50 lbs or less The new bag’s mean breaking strength is not known and is in question, but it is hoped it is stronger than the current one

Example 9.1: Trash Bag Case Continued
The alternative hypothesis Ha is that the new bag has a mean breaking strength that exceeds 50 lbs One-sided, “greater than” alternative H0: µ ≤ versus Ha: µ > 50

Example 9.2: Payment Time Case
With new billing system, the mean bill paying time µ is hoped to be less than 19.5 days Alternative hypothesis Ha is the new billing system has a mean payment time less than 19.5 days With old billing system, the mean bill paying time µ was close to but not less than 39 days The null hypothesis H0 is that the new billing system has a mean payment time close to but not less than 39 days One-sided, “less than” alternative H0:   19.5 versus Ha:  < 19.5

Example 9.3: The Valentine’s Day Chocolate Case
They have designed a new box They hope sales increase by 10 percent Last year’s sales were 330 boxes The null hypothesis will be that sales this year will not be 330 boxes Two-sided, “not equal to” alternative H0:  = 330 versus Ha:  ≠ 330

Types of Hypotheses One-Sided, “Greater Than” Alternative
H0:   m0 vs. Ha:  > m0 One-Sided, “Less Than” Alternative H0 :   m0 vs. Ha :  < m0 Two-Sided, “Not Equal To” Alternative H0 :  = m0 vs. Ha :   m0 where m0 is a given constant value (with the appropriate units) that is a comparative value

Types of Decisions As a result of testing H0 vs. Ha, will have to decide either of the following decisions for the null hypothesis H0: Do not reject H0 A weaker statement than “accepting H0” But you are rejecting the alternative Ha Or reject H0 A weaker statement than “accepting Ha”

Test Statistic To “test” H0 vs. Ha, use the “test statistic” where m0 is the given value (often the “claimed to be true”) and x is the sample mean z measures the distance between m0 and x on the sampling distribution of the sample mean If the population is normal or n is large*, the test statistic z follows a normal distribution * n ≥ 30, by the Central Limit Theorem

Test Statistic and Trash Bag Case
With µ0 = 50, use the test statistic: If z ≤ 0, then x ≤ µ0 and there is no evidence to support rejecting H0 in favor of Ha The point estimate x indicates µ is probably less than 50 If z > 0, then x > µ0 and there is evidence to support rejecting H0 in favor of Ha The point estimate x indicates that µ is probably greater than 50 The larger z (the farther x is above µ), the stronger the evidence to support rejecting H0 in favor of Ha

Type I and Type II Errors
Type I Error: Rejecting H0 when it is true Type II Error: Failing to reject H0 when it is false

Error Probabilities Type I Error: Rejecting H0 when it is true
 is the probability of making a Type I error 1 –  is the probability of not making a Type I error Type II Error: Failing to reject H0 when it is false β is the probability of making a Type II error 1 – β is the probability of not making a Type II error

Error Probabilities State of Nature Conclusion H0 True H0 False
Reject H0  1-  Do not Reject H0 1-β β

Typical Values Usually set  to a low value
So there is a small chance of rejecting a true H0 Typically,  = 0.05 Strong evidence is required to reject H0 Usually choose  between 0.01 and 0.05  = 0.01 requires very strong evidence is to reject H0 Tradeoff between  and β For fixed sample size, the lower , the higher β And the higher , the lower β

z Tests about a Population Mean: σ Known
Test hypotheses about a population mean using the normal distribution Called z tests Require that the true value of the population standard deviation σ is known In most real-world situations, σ is not known But often is estimated from s of a single sample When σ is unknown, test hypotheses about a population mean using the t distribution Here, assume that we know σ Also use a “rejection rule”

Steps in Testing a “Greater Than” Alternative
State the null and alternative hypotheses Specify the significance level a Select the test statistic Determine the rejection rule for deciding whether or not to reject H0 Collect the sample data and calculate the value of the test statistic Decide whether to reject H0 by using the test statistic and the rejection rule Interpret the statistical results in managerial terms and assess their practical importance

Steps in Testing a “Greater Than” Alternative in Trash Bag Case #1
State the null and alternative hypotheses H0:   50 Ha:  > 50 where μ is the mean breaking strength of the new bag Specify the significance level   = 0.05

Select the test statistic Use the test statistic A positive value of this this test statistic results from a sample mean that is greater than 50 lbs Which provides evidence against H0 in favor of Ha

Determine the rejection rule for deciding whether or not to reject H0 To decide how large the test statistic must be to reject H0 by setting the probability of a Type I error to , do the following: The probability  is the area in the right-hand tail of the standard normal curve Use the normal table to find the point z (called the rejection or critical point) z is the point under the standard normal curve that gives a right-hand tail area equal to  Since  = 0.05 in the trash bag case, the rejection point is z = z0.05 = 1.645

Continued Reject H0 in favor of Ha if the test statistic z is greater than the rejection point z This is the rejection rule In the trash bag case, the rejection rule is to reject H0 if the calculated test statistic z is > 1.645

Collect the sample data and calculate the value of the test statistic In the trash bag case, assume that σ is known and σ = 1.65 lbs For a sample of n = 40, x = lbs. Then

Decide whether to reject H0 by using the test statistic and the rejection rule Compare the value of the test statistic to the rejection point according to the rejection rule In the trash bag case, z = 2.20 is greater than z0.05 = 1.645 Therefore reject H0: μ ≤ 50 in favor of Ha: μ > 50 at the 0.05 significance level Have rejected H0 by using a test that allows only a 5% chance of wrongly rejecting H0 This result is “statistically significant” at the 0.05 level

Interpret the statistical results in managerial terms and assess their practical importance Can conclude that the mean breaking strength of the new bag exceeds 50 lbs

Testing H0:   50 versus Ha:  > 50

Effect of  At  = 0.01, the rejection point is z0.01 = 2.33
In the trash example, the test statistic z = 2.20 is < z0.01 = 2.33 Therefore, cannot reject H0 in favor of Ha at the  = 0.01 significance level This is the opposite conclusion reached with =0.05 So, the smaller we set , the larger is the rejection point, and the stronger is the statistical evidence that is required to reject the null hypothesis H0

The p-Value The p-value or the observed level of significance is the probability of the obtaining the sample results if the null hypothesis H0 is true The p-value is used to measure the weight of the evidence against the null hypothesis Sample results that are not likely if H0 is true have a low p-value and are evidence that H0 is not true The p-value is the smallest value of  for which we can reject H0 The p-value is an alternative to testing with a z test statistic

The p-Value for “Greater Than” Alternative
For a greater than alternative, the p-value is the probability of observing a value of the test statistic greater than or equal to z when H0 is true Use p-value as an alternative to testing a greater than alternative with a z test statistic

The p-value for “Greater Than” Example: Trash Bag Case
Testing H0: µ ≤ 50 versus Ha: µ > 50 using rejection points in (a) and p-value in (b) Have to modify steps 4, 5, and 6 of the previous procedure to use p-values

Steps Using a p-value to Test a “Greater Than” Alternative
Collect the sample data and compute the value of the test statistic In the trash bag case, the value of the test statistic was calculated to be z = 2.20 Calculate the p-value by corresponding to the test statistic value In the trash bag case, the area under the standard normal curve in the right-hand tail to the right of the test statistic value z = 2.20 The area is 0.5 – = The p-value is

Steps Using a p-value to Test a “Greater Than” Alternative Continued
If H0 is true, the probability is of obtaining a sample whose mean is lbs or higher This is so low as to be evidence that H0 is false and should be rejected Reject H0 if the p-value is less than  In the trash bag case, a was set to 0.05 The calculated p-value of is <  = 0.05 This implies that the test statistic z = 2.20 is greater than the rejection point z0.05 = 1.645 Therefore reject H0 at the  = 0.05 significance level

Steps in Testing a “Less Than” Alternative

Steps in Testing a “Less Than” Alternative in Payment Time Case #1
State the null and alternative hypotheses In the payment time case, H0:  ≥ 19.5 vs. Ha:  < 19.5, where  is the mean bill payment time (in days) Specify the significance level  In the payment time case, set  = 0.01

Select the test statistic In the payment time case, use the test statistic A negative value of this this test statistic results from a sample mean that is less than 19.5 days Which provides evidence against H0 in favor of Ha

Determine the rejection rule for deciding whether or not to reject H0 To decide how much less than 0 the test statistic must be to reject H0 by setting the probability of a Type I error to , do the following: The probability  is the area in the left-hand tail of the standard normal curve Use normal table to find the rejection point –z –z is the negative of z –z is the point on the horizontal axis under the standard normal curve that gives a left-hand tail area equal to 

Continued Since  = 0.01 in the payment time case, the rejection point is –z = –z0.01 = –2.33 Reject H0 in favor of Ha if the test statistic z is calculated to be less than the rejection point –z This is the rejection rule In the payment time case, the rejection rule is to reject H0 if the calculated test statistic –z is less than –2.33

Collect the sample data and calculate the value of the test statistic In the payment time case, assume that σ is known and σ = 4.2 days For a sample of n=65, x = days:

Decide whether to reject H0 by using the test statistic and the rejection rule Compare the value of the test statistic to the rejection point according to the rejection rule In the payment time case, z = –2.67 is less than z0.01 = –2.33 Therefore reject H0: μ ≥ 19.5 in favor of Ha: μ < 19.5 at the 0.01 significance level Have rejected H0 by using a test that allows only a 1% chance of wrongly rejecting H0 This result is “statistically significant” at the 0.01 level

Interpret the statistical results in managerial terms and assess their practical importance Can conclude that the mean bill payment time of the new billing system is less than 19.5 days

Steps in Testing a “Less Than” Alternative in Payment Time Case
Testing H0: µ ≥ 19.5 versus Ha: µ < 19.5 for  = 0.01

The p-value for “Less Than”
Have to modify steps 4, 5, and 6 of the previous procedure to use p-values

Steps Using a p-value to Test a “Less Than” Alternative
(Steps 1–3 are the same) Collect the sample data and compute the value of the test statistic In the payment time case, the value of the test statistic was calculated to be z = –2.67 Calculate the p-value by corresponding to the test statistic value In the payment time case, the area under the standard normal curve in the left-hand tail to the left of the test statistic z = –2.67 The area is 0.5 – = The p-value is

Steps Using a p-value to Test a “Less Than” Alternative Continued
If H0 is true, then the probability is of obtaining a sample whose mean is as low as days or lower This is so low as to be evidence that H0 is false and should be rejected Reject H0 if the p-value is less than  In the payment time case,  was 0.01 The calculated p-value of is <  = 0.01 This implies that the test statistic z = –2.67 is less than the rejection point –z0.01 = –2.33 Therefore, reject H0 at the  = 0.01 significance level

Summary of Testing a One-Sided Alternative Using a Test Statistic
If the population is normal and s is known*, we can reject H0:  = 0 at the  level of significance (probability of Type I error equal to ) if and only if the appropriate rejection point rule holds Test Statistic Alternative Reject H0 if: Ha : m > m0 z > za Ha : m < m0 z < za * If  unknown and n is very large (n > 120), estimate  by s

Summary of Testing a One-Sided Alternative Using a p-value
If the population is normal and s is known*, we can reject H0:  = 0 at the  level of significance (probability of Type I error equal to ) if and only if the corresponding p-value is less than the specified  Test Statistic Alternative Reject H0 if: Ha : m > m0 p > za Ha : m < m0 p < za (1) a is the area under the standard normal curve to the right of za a is the area under the standard normal curve to the left of -za * If  unknown and n is very large (n > 120), estimate  by s

Steps in Testing a “Not Equal To” Alternative

Steps in Testing a “Not Equal To” Alternative in Valentine Day Case #1
State null and alternative hypotheses In the Valentine Day case, H0:  = 330 vs. Ha:  ≠ 330 Specify the significance level  In the Valentine Day case, set  = 0.05 Select the test statistic In this case, use the test statistic

Continued A positive value of this test statistic results from a sample mean that is greater than 330 Which provides evidence against H0 and for Ha A negative value of this test statistic results from a sample mean that is less than 330 A very small value close to 0 (either slightly positive or slightly negative) of this test statistic results from a sample mean that is nearly 330 Which provides evidence in favor of H0 and against Ha

Determine the rejection rule for deciding whether or not to reject H0 To decide how different the test statistic must be from zero (positive or negative) to reject H0 in favor of Ha by setting the probability of a Type I error to , do the following: Divide the  in half to find /2; /2 is the tail area in both tails of the standard normal curve Under the standard normal curve, the probability /2 is the area in the right-hand tail and probability /2 is the area in the left-hand tail

Continued Use the normal table to find the rejection points z/2 and –z/2 z/2 is the point on the horizontal axis under the standard normal curve that gives a right-hand tail area equal to /2 –z/2 is the point on the horizontal axis under the standard normal curve that gives a left-hand tail area equal to /2

Continued Because  = 0.05, /2=0.025 The area under the standard normal to the right of the mean is 0.5 – = 0.475 From Table A.3, the area is for z = 1.96 Rejection points are z=1.96,–z=– 1.96 Reject H0 in favor of Ha if the test statistic z satisfies either: z greater than the rejection point z/2, or –z less than the rejection point –z/2 This is the rejection rule

Collect the sample data and calculate the value of the test statistic In the Valentine Day case, assume that σ is known and σ = 40 For a sample of n = 100, x = 326 Then

Decide whether to reject H0 by using the test statistic and the rejection rule Compare the value of the test statistic to the rejection point according to the rejection rule In this case, – z = –1.00 is < – z0.025 = –1.96 Therefore cannot reject H0: µ = 330 in favor of Ha: µ ≠ 330 at the 0.05 significance level Have not rejected H0 by using a test that allows only a 5% chance of wrongly rejecting H0

Interpret the statistical results in managerial terms and assess their practical importance Cannot conclude that the mean order quantity this year of the Valentine Day box at large retail stores will differ from 330 boxes

Steps Using a p-value to Test a “Not Equal To” Alternative
(Steps 1–3 are the same) Collect the sample data and compute the value of the test statistic In the Valentine Day case, the value of the test statistic was calculated to be z = –1.00 Calculate the p-value by corresponding to the test statistic value In the Valentine Day case, the area under the standard normal curve in the left-hand tail to the left of the test statistic value z = –1.00 The area is The p-value is · 2 =

Steps Using a p-value to Test a “Not Equal To” Alternative Continued
That is, if H0 is true, then the probability is of obtaining a sample whose mean is at least as extreme as 326 This probability is not so low as to be evidence that H0 is false and should be rejected Reject H0 if the p-value is less than a In the Valentine Day case,  was 0.05 Calculated p-value of is greater than  This implies that the test statistic z = –1.00 is greater than the rejection point –z0.025 = –1.96 Therefore do not reject H0 at the  = 0.05 significance level

Testing H0: µ = 330 Versus Ha: µ ≠ 330 by Using Critical Values and the p-Value

Summary of Testing a Two-Sided Not Equal Alternative Using a p-value
If the population is normal and s is known*, we can reject H0:  = 0 at the  level of significance (probability of Type I error equal to ) if and only if: Test Statistic Alternative Reject H0 if: Ha : m > m0 z > za/2 or p < a (1) z > –za/2 or p < a (2) (1) a is the area under the standard normal curve to the right of za a is the area under the standard normal curve to the left of -za * If  unknown and n is very large (n > 120), estimate  by s

Weight of Evidence Against the Null
Calculate the test statistic and the corresponding p-value Rate the strength of the conclusion about the null hypothesis H0 according to these rules: If p < 0.10, there is some evidence to reject H0 If p < 0.05, there is strong evidence to reject H0 If p < 0.01, there is very strong evidence to reject H0 If p < 0.001, there is extremely strong evidence to reject H0

t Tests about a Population Mean: σ Unknown
Suppose the population being sampled is normally distributed The population standard deviation σ is unknown, as is the usual situation If the population standard deviation σ is unknown, then it will have to estimated from a sample standard deviation s Under these two conditions, have to use the t distribution to test hypotheses

Defining the t Random Variable: σ Unknown
Define a new random variable t: with the definition of symbols as before The sampling distribution of this random variable is a t distribution with n – 1 degrees of freedom

Defining the t Statistic: σ Unknown
Let x be the mean of a sample of size n with standard deviation s Also, µ0 is the claimed value of the population mean Define a new test statistic If the population being sampled is normal, and s is used to estimate σ, then … The sampling distribution of the t statistic is a t distribution with n – 1 degrees of freedom

t Tests about a Population Mean: σ Unknown
Reject H0: µ = µ0 in favor of a particular alternative hypothesis Ha at the a level of significance if and only if the appropriate rejection point rule or, equivalently, the corresponding p-value is less than  We have the following rules …

t Tests about a Population Mean: σ Unknown Continued
Alternative Reject H0 if: p-value Ha: µ > µ0 t > ta Area under t distribution to right of t Ha: µ < µ0 t < –ta Area under t distribution to left of –t Ha: µ  µ0 |t| > t a/2 * Twice area under t distribution to right of |t| t, t/2, and p-values are based on n – 1 degrees of freedom (for a sample of size n) * either t > t/2 or t < –t/2

Example 9.4 #1 The current mean credit card interest rate is as it was in 1991, at the rate of 18.8% This is the null hypothesis H0: µ = 18.8 The alternative to be tested is that the current mean interest rate is not as it was back in 1991, but in fact has decreased This is the alternative hypothesis Ha: µ  18.8 Test at the  = 0.05 level of significance If H0 can be rejected at the 5% level, then conclude that the mean rate now is less than 18.8% rate charged in 1991

Example 9.4 #2 Randomly select n = 15 credit cards
 = 0.05 Then df = n – 1 = 14 Assume the population of the rates of all cards is normal

Example 9.4 #3 The rejection rule is reject H0: m = 18.8 against Ha: m < 18.8 if t < ta For a = 0.05 and with df = 14, ta = t0.05 =1.761 Calculate the value of the t statistic:

Example 9.4 #4

Example 9.4 #5 Because t = 4.97 < t0.05 = 1.761, reject H0
Therefore, conclude at 5% significance level that the mean interest rate is lower than it was in 1991 In fact it is µ0 – x = 18.8 – = 1.973% lower

p-value for Example 9.4 The p-value is the left-hand tail area under the t curve with df = 14 to the left of t = 4.97 This t value is off the t table, so use statistics software on a computer to calculate the p-value p = This says that if the claimed mean of 18.8 % is true, then there is only a 0.01% chance of randomly selecting a sample of 15 credit whose mean rate would be as low as % or lower This p-value is less than 0.05, 0.01, and 0.001 Reject H0 at the 0.05, 0.01, significance levels

Computer Software Output for Example 9.4

Hypothesis Tests about a Population Proportion
If the sample size n is large, we can reject H0: ρ = ρ0 at the  level of significance (probability of Type I error equal to ) if and only if the appropriate rejection point condition holds or, equivalently, if the corresponding p-value is less than  We have the following rules …

A Large Sample Test About a Population Proportion
Alternative Reject H0 if: p-value Ha: ρ > ρ0 z > za Area under t distribution to right of z Ha: ρ < ρ0 z < –za Area under t distribution to left of –z Ha: ρ  ρ0 |z| > z a/2 * Twice area under t distribution to right of |z| Where the test statistics is * either z > z/2 or z < –z/2

Example 9.7 Using Phantol, proportion of patients with reduced severity and duration of viral infections Test H0: p ≤ 0.70 versus Ha: p > 0.70 using rejection points and p-values

Example 9.7 Continued Comparing values:
z = 2.65 > z0.05 = z = 2.65 > z0.01 = 2.33 z = 2.65 < z0.001 = 3.09 Reject H0 at the 0.05 and 0.01 significance levels but do not reject at the level p-value = P(z > 2.65) = (0.5 – ) = but p = <  = 0.05 p = <  = 0.01 p = >  = 0.001 Again reject H0 at the 0.05 and 0.01 levels but do not reject at the level

Type II Error Probabilities and Sample Size Determination
Want the probability β of not rejecting a false null hypothesis That is, want the probability β of committing a Type II error 1 - β is called the power of the test

Example: Type II Error Probabilities #1
The claim: cans of coffee contain at least 3 lbs of coffee Want to statistically test if the mean amount of coffee µ in all cans is at least 3 pounds Test H0: µ ≥ 3 versus Ha: µ < 3 Sample: n = 35, x = lbs, σ = lbs Note that σ is assumed to be known here Test at the  = 0.05 significance level

A Type II error is not rejecting H0: µ ≥ 3 when H0 is false But any value of µ less than 3 makes H0 false So, there is a different Type II error probability for each value of µ less than 3 On the next two slides, calculate the probability of not rejecting H0 when in fact µ actual equals 2.995

Suppose that H0 is false and m = lbs

Suppose that H0 is false and m = lbs continued

Example: How Type II Error Varies Against Alternatives
Testing H0:   3 vs. Ha:  < 3 (Amount of Coffee in 3-Pound Can) , Probability of Type II Error ( = 0.05) Given  = 2.995, b = 0.557 Given  = 2.990, b = Given  = 2.985, b < 0.001 In general, the lower the actual m is from the claimed value, the lower b, so the lower the chance of making a Type II error

Calculating β Assume that the sampled population is normally distributed, or that a large sample is taken Test… H0: µ = µ0 vs Ha: µ < µ0 or Ha: µ > µ0 or Ha: µ ≠ µ0 Want to make the probability of a Type I error equal to  and randomly select a sample of size n

Calculating β Continued
The probability β of a Type II error corresponding to the alternative value µa for µ is equal to the area under the standard normal curve to the left of Here z* equals z if the alternative hypothesis is one-sided (µ < µ0 or µ > µ0) Also z* ≠ z/2 if the alternative hypothesis is two-sided (µ ≠ µ0)

Sample Size Assume that the sampled population is normally distributed, or that a large sample is taken Test H0: m = m0 vs. Ha: m < m0 or Ha: m > m0 or Ha: m ≠ m0 Want to make the probability of a Type I error equal to a and the probability of a Type II error corresponding to the alternative value ma for m equal to b

Sample Size Continued Then take a sample of size:
Here z* equals za, if the alternative hypothesis is one-sided (m < m0 or m > m0) and z* equals za/2 if the alternative hypothesis is two-sided (m ≠ m0) Also zb is the point on the scale of the standard normal curve that gives a right-hand tail area equal to b

Example 9.10: Coffee Fill Example
Wish to test H0: µ ≥ 3 versus Ha: µ < 3  = 0.05 β = 0.05 for the alternative value µa = 2.995 z* = z = z0.05 = since Ha is one-sided

The Chi-Square Distribution
The chi-square ² distribution depends on the number of degrees of freedom See Table A.17 of Appendix A A chi-square point ² is the point under a chi-square distribution that gives right-hand tail area 

A Portion of the Chi-Square Table

Statistical Inference for Population Variance
If s2 is the variance of a random sample of n measurements from a normal population with variance σ2 The sampling distribution of the statistic (n - 1) s2 / σ2 is a chi-square distribution with (n – 1) degrees of freedom Can calculate confidence interval and perform hypothesis testing

Confidence Interval for Population Variance
All chi-square points are based on n – 1 degrees of freedom

Hypothesis Testing for Population Variance
Test of H0: σ2 = σ20 Test Statistic: Reject H0 in favor of Ha: σ2 > σ20 if 2 > 2 Ha: σ2 < σ20 if 2 < 21- Ha: σ2 ≠ σ20 if 2 > 2 or 2 < 21-

Selecting an Appropriate Test Statistic for a Test about a Population Mean

Chapter 9 Hypothesis Testing.

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