AP Statistics Curriculum 2007 Hypothesis Basics - Revision history

Tdlee: Text replacement - "{{translate|pageName=http://wiki.stat.ucla.edu/socr/" to ""{{translate|pageName=http://wiki.socr.umich.edu/"

2020-03-03T17:17:50Z

Text replacement - "{{translate|pageName=http://wiki.stat.ucla.edu/socr/" to ""{{translate|pageName=http://wiki.socr.umich.edu/"

Dinov: /* Type I Error, Type II Error and Power */

2014-09-10T13:33:50Z

‎Type I Error, Type II Error and Power

Dinov: /* Type I Error, Type II Error and Power */

2014-08-05T19:00:19Z

‎Type I Error, Type II Error and Power

Dinov: /* Type I Error, Type II Error and Power */

2014-08-05T18:57:36Z

‎Type I Error, Type II Error and Power

IvoDinov: /* Example 2: Sodium content in hot-dogs */

2013-05-22T15:54:10Z

‎Example 2: Sodium content in hot-dogs

IvoDinov: /* Type I Error, Type II Error and Power */

2013-05-22T15:53:43Z

‎Type I Error, Type II Error and Power

IvoDinov: /* Example 3: Rapid testing in strep-throat */

2012-02-29T16:57:28Z

‎Example 3: Rapid testing in strep-throat

IvoDinov: /* Example 3: Rapid testing in strep-throat */

2012-02-29T16:55:37Z

‎Example 3: Rapid testing in strep-throat

Jenny: /* Example 2: Sodium content in hot-dogs */

2010-06-29T18:06:59Z

‎Example 2: Sodium content in hot-dogs

Jenny at 20:40, 28 June 2010

2010-06-28T20:40:11Z

← Older revision		Revision as of 17:17, 3 March 2020
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	* SOCR Home page: http://www.socr.ucla.edu		* SOCR Home page: http://www.socr.ucla.edu

−	{{translate\|pageName=http://wiki.~~stat~~.~~ucla~~.edu~~/socr~~/index.php?title=AP_Statistics_Curriculum_2007_Hypothesis_Basics}}	+	"{{translate\|pageName=http://wiki.socr.umich.edu/index.php?title=AP_Statistics_Curriculum_2007_Hypothesis_Basics}}

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 ** '''Sensitivity''' is a measure of how well a test correctly identifies a condition, whether this is medical screening tests picking up on a disease, or quality control in factories deciding if a new product is good enough to be sold. $Sensitivity=\frac{TP}{TP+FN} =\frac{0.00475}{0.00475+ 0.00025}= 0.95.$
 ** '''False Negative Rate (β)'''= \(\frac{FN}{FN+TP} = \frac{0.00025}{0.00025+0.00475}=0.05 \)= 1 - Sensitivity.
-** '''Power''' = 1 − β= 0.99975. Power $= P(\mbox{reject null hypothesis} | \mbox{null hypothesis is false})$. For example, see [[Power_Analysis_for_Normal_Distribution]].
+** '''Power''' = 1 − β= 0.95. Power $= P(\mbox{reject null hypothesis} | \mbox{null hypothesis is false})$. For example, see [[Power_Analysis_for_Normal_Distribution]].
 ===Example 2: Sodium content in hot-dogs===

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 ** A '''Specificity''' of 100% means that the test recognizes all healthy individuals as (normal) healthy. The maximum is trivially achieved by a test that claims everybody is healthy regardless of the true condition. Therefore, the specificity alone does not tell us how well the test recognizes positive cases.
 ** '''False positive rate (α)'''= \(\frac{FP}{FP+TN} = \frac{0.00995}{0.00995 + 0.98505}=0.01 \)= 1 - Specificity.
-** '''Sensitivity''' is a measure of how well a test correctly identifies a condition, whether this is medical screening tests picking up on a disease, or quality control in factories deciding if a new product is good enough to be sold.
+** '''Sensitivity''' is a measure of how well a test correctly identifies a condition, whether this is medical screening tests picking up on a disease, or quality control in factories deciding if a new product is good enough to be sold. $Sensitivity=\frac{TP}{TP+FN} =\frac{0.00475}{0.00475+ 0.00025}= 0.95.$
 ** '''False Negative Rate (β)'''= \(\frac{FN}{FN+TP} = \frac{0.00025}{0.00025+0.00475}=0.05 \)= 1 - Sensitivity.
 ** '''Power''' = 1 − β= 0.99975. Power $= P(\mbox{reject null hypothesis} | \mbox{null hypothesis is false})$. For example, see [[Power_Analysis_for_Normal_Distribution]].

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 ** '''Sensitivity''' is a measure of how well a test correctly identifies a condition, whether this is medical screening tests picking up on a disease, or quality control in factories deciding if a new product is good enough to be sold.
 ** '''False Negative Rate (β)'''= \(\frac{FN}{FN+TP} = \frac{0.00025}{0.00025+0.00475}=0.05 \)= 1 - Sensitivity.
-** '''Power''' = 1 − β= 0.99975, see [[Power_Analysis_for_Normal_Distribution]].
+** '''Power''' = 1 − β= 0.99975. Power $= P(\mbox{reject null hypothesis} | \mbox{null hypothesis is false})$. For example, see [[Power_Analysis_for_Normal_Distribution]].
 ===Example 2: Sodium content in hot-dogs===

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 ===Example 2: Sodium content in hot-dogs===
 Use the [[SOCR_012708_ID_Data_HotDogs |Hot-dog dataset]] to see if there are statistically significant differences in the sodium content of the poultry vs. meat hotdogs.
-* Formulate Hypotheses: <math>H_o: \mu_p = \mu_m</math> vs. <math>H_1: \mu_p \not= \mu_m</math>, where <math>\mu_p, \mu_m</math> represent the mean sodium content in poultry and mean hotdogs.
+* Formulate Hypotheses: \(H_o: \mu_p = \mu_m\) vs. \(H_1: \mu_p \not= \mu_m\), where <math>\mu_p, \mu_m</math> represent the mean sodium content in poultry and mean hotdogs.
 * Plug in the data in [http://socr.stat.ucla.edu/htmls/SOCR_Analyses.html SOCR Analyses] under the [[SOCR_EduMaterials_AnalysisActivities_TwoIndepTU |Two Independent Sample T-Test (Unpooled)]] will generate results as shown in the figure below (Two-Sided P-Value (Unpooled) = 0.196, which does not provide strong evidence to reject the null hypothesis that the two types of hot-dogs have the same mean sodium content).
 <center>[[Image:SOCR_EBook_Dinov_Hypothesis_020508_Fig1.jpg|600px]]</center>

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 The study found that the rapid strep test in this setting showed no difference in specificity (0.96 vs. 0.98). Hence, the assertion that rapid antigen testing had higher false-positive rates in those with recent infection was not confirmed. It also found that in patients who had recent streptococcal pharyngitis, the rapid strep test appears to be more reliable (sensitivity 0.91 vs 0.70, P < .001) than in those patients who had not had recent streptococcal pharyngitis. These findings indicated that the rapid strep test is both sensitive and specific in the setting of recent group A <math>\beta</math>-hemolytic streptococcal pharyngitis, and its use might allow earlier treatment in this subgroup of patients.
-Table 1. Sensitivity and Specificity of Laboratory Culture and Rapid Strep Test in P''atients With Recently Treated Cases of Streptococcal Pharyngitis''.
+Table 1. Sensitivity and Specificity of Laboratory Culture and Rapid Strep Test in ''Patients With Recently Treated Cases of Streptococcal Pharyngitis'' (N=443).
 <center>
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 </center>
-Table 2. Sensitivity and Specificity of Laboratory Culture and Rapid Strep Test in ''Patients With No Recently Treated Cases of Streptococcal Pharyngitis''.
+Table 2. Sensitivity and Specificity of Laboratory Culture and Rapid Strep Test in ''Patients With No Recently Treated Cases of Streptococcal Pharyngitis'' (N=232).
 <center>

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 Study used 443 patients who had clinical pharyngitis diagnosed as group A <math>\beta</math>-hemolytic streptococcus infection in the past 28 days and compared them with 232 control patients who had symptoms of pharyngitis but no recent diagnosis of streptococcal pharyngitis. The aim was narrowly focused to compare the rapid strep test with the culture method used in clinical practice.
-The study found that the rapid strep test in this setting showed no difference in specificity (0.96 vs. 0.98). Hence, the assertion that rapid antigen testing had higher false-positive rates in those with recent infection was not confirmed. It also found that in patients who had recent streptococcal pharyngitis, the rapid strep test appears to be more reliable (0.91 vs 0.70, P < .001) than in those patients who had not had recent streptococcal pharyngitis. These findings indicated that the rapid strep test is both sensitive and specific in the setting of recent group A <math>\beta</math>-hemolytic streptococcal pharyngitis, and its use might allow earlier treatment in this subgroup of patients.
+The study found that the rapid strep test in this setting showed no difference in specificity (0.96 vs. 0.98). Hence, the assertion that rapid antigen testing had higher false-positive rates in those with recent infection was not confirmed. It also found that in patients who had recent streptococcal pharyngitis, the rapid strep test appears to be more reliable (sensitivity 0.91 vs 0.70, P < .001) than in those patients who had not had recent streptococcal pharyngitis. These findings indicated that the rapid strep test is both sensitive and specific in the setting of recent group A <math>\beta</math>-hemolytic streptococcal pharyngitis, and its use might allow earlier treatment in this subgroup of patients.
 Table 1. Sensitivity and Specificity of Laboratory Culture and Rapid Strep Test in P''atients With Recently Treated Cases of Streptococcal Pharyngitis''.

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 === Fundamentals of Hypothesis Testing===
-A (statistical) '''Hypothesis Test''' is a method of making statistical decisions about populations or processes based on experimental data.  Hypothesis testing just answers the question of ''how well the findings fit the possibility that chance alone might be responsible for the observed discrepancy between the theoretical model and the empirical observations''. This is accomplished by asking and answering a hypothetical question - what is the likelihood of the observed summary statistics of interest, if the data did come from the distribution specified by the null-hypothesis. One use of hypothesis-testing is deciding whether experimental results contain enough information to cast doubt on conventional wisdom.
+A (statistical) '''Hypothesis Test''' is a method of making statistical decisions about populations or processes based on experimental data.  Hypothesis testing just answers the question of ''how well the findings fit the possibility that the chance alone might be responsible for the observed discrepancy between the theoretical model and the empirical observations''. This is accomplished by asking and answering a hypothetical question. What is the likelihood of the observed summary statistics of interest, if the data did come from the distribution specified by the null-hypothesis? One use of hypothesis-testing is deciding whether experimental results contain enough information to cast doubt on conventional wisdom.
 * Example: Consider determining whether a suitcase contains some radioactive material. Placed under a [http://en.wikipedia.org/wiki/Geiger_counter Geiger counter], the suitcase produces 10 clicks (counts) per minute. The '''null hypothesis''' is that there is no radioactive material in the suitcase and that all measured counts are due to ambient radioactivity typical of the surrounding air and harmless objects in a suitcase. We can then calculate how likely it is that the null hypothesis produces 10 counts per minute. If it is likely, for example if the null hypothesis predicts on average 9 counts per minute, we say that the suitcase is compatible with the null hypothesis (which does not imply that there is no radioactive material, we just can't determine from the 1-minute sample we took using this specific method!); On the other hand, if the null hypothesis predicts for example 1 count per minute, then the suitcase is not compatible with the null hypothesis and there must be other factors responsible to produce the increased radioactive counts.
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 Alternatively, the null hypothesis can postulate that the two samples are drawn from the same population, so that the [[AP_Statistics_Curriculum_2007#Chapter_II:_Describing.2C_Exploring.2C_and_Comparing_Data | center, variance and shape of the distributions]] are equal.
-Formulation of the null hypothesis is a vital step in testing statistical significance.  Having formulated such a hypothesis, one can establish the probability of observing the obtained data from the prediction of the null hypothesis, if the null hypothesis is true. That probability is what is commonly called the ''significance level'' of the results.
+Formulation of the null hypothesis is a vital step in testing statistical significance.  Having formulated such a hypothesis, one can establish the probability of observing the obtained data from the prediction of the null hypothesis, if the null hypothesis is true. That probability is what commonly called the ''significance level'' of the results.
 In many scientific experimental designs we predict that a particular factor will produce an effect on our dependent variable — this is our alternative hypothesis. We then consider how often we would expect to observe our experimental results, or results even more extreme, if we were to take many samples from a population where there was no effect (i.e. we test against our null hypothesis). If we find that this happens rarely (up to, say, 5% of the time), we can conclude that our results support our experimental prediction — we reject our null hypothesis.
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 Study used 443 patients who had clinical pharyngitis diagnosed as group A <math>\beta</math>-hemolytic streptococcus infection in the past 28 days and compared them with 232 control patients who had symptoms of pharyngitis but no recent diagnosis of streptococcal pharyngitis. The aim was narrowly focused to compare the rapid strep test with the culture method used in clinical practice.
-The study found that the rapid strep test in this setting showed no difference in specificity (0.96 vs. 0.98); hence, the assertion that rapid antigen testing had higher false-positive rates in those with recent infection was not confirmed. It also found that in patients who had recent streptococcal pharyngitis, the rapid strep test appears to be more reliable (0.91 vs 0.70, P < .001) than in those patients who had not had recent streptococcal pharyngitis. These findings indicated that the rapid strep test is both sensitive and specific in the setting of recent group A <math>\beta</math>-hemolytic streptococcal pharyngitis, and its use might allow earlier treatment in this subgroup of patients.
+The study found that the rapid strep test in this setting showed no difference in specificity (0.96 vs. 0.98). Hence, the assertion that rapid antigen testing had higher false-positive rates in those with recent infection was not confirmed. It also found that in patients who had recent streptococcal pharyngitis, the rapid strep test appears to be more reliable (0.91 vs 0.70, P < .001) than in those patients who had not had recent streptococcal pharyngitis. These findings indicated that the rapid strep test is both sensitive and specific in the setting of recent group A <math>\beta</math>-hemolytic streptococcal pharyngitis, and its use might allow earlier treatment in this subgroup of patients.
 Table 1. Sensitivity and Specificity of Laboratory Culture and Rapid Strep Test in P''atients With Recently Treated Cases of Streptococcal Pharyngitis''.