document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1163235>Question 1163235</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The screening process for detecting a rare disease is not perfect. Researchers have developed a blood test that is considered fairly reliable. It gives a positive reaction in 98.2% of the people who have that disease. However, it erroneously gives a positive reaction in 1.3% of the people who do not have the disease. \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "a. Calculate the probability of a Type I error and round your answer to 3 decimal places.\r<BR>\n" );
document.write( "\n" );
document.write( "b. Calculate the probability of a Type II error and round your answer to 3 decimal places. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #854390 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2264)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=854390\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> In hypothesis testing, a Type I error and a Type II error represent the two ways a test can give an incorrect result. To solve this, we first define our \"hypotheses\" in the context of the disease:\r<BR>\n" );
document.write( "\n" );
document.write( "* **$H_0$ (Null Hypothesis):** The person does **not** have the disease.<BR>\n" );
document.write( "* **$H_A$ (Alternative Hypothesis):** The person **has** the disease.\r<BR>\n" );
document.write( "\n" );
document.write( "---\r<BR>\n" );
document.write( "\n" );
document.write( "### a. Calculate the probability of a Type I error<BR>\n" );
document.write( "A **Type I error** (often denoted as $\alpha$) occurs when we reject the null hypothesis even though it is true. In medical terms, this is a **False Positive**.<BR>\n" );
document.write( "* This happens when the test says a person **has** the disease, but they actually **do not**.<BR>\n" );
document.write( "* The problem states the test erroneously gives a positive reaction in **1.3%** of people who do not have the disease.\r<BR>\n" );
document.write( "\n" );
document.write( "**Calculation:**<BR>\n" );
document.write( "$$P(\text{Type I Error}) = 1.3\% = 0.013$$\r<BR>\n" );
document.write( "\n" );
document.write( "> **Probability of a Type I error: 0.013**\r<BR>\n" );
document.write( "\n" );
document.write( "---\r<BR>\n" );
document.write( "\n" );
document.write( "### b. Calculate the probability of a Type II error<BR>\n" );
document.write( "A **Type II error** (often denoted as $\beta$) occurs when we fail to reject the null hypothesis even though it is false. In medical terms, this is a **False Negative**.<BR>\n" );
document.write( "* This happens when the test says a person **does not** have the disease, but they actually **do**.<BR>\n" );
document.write( "* The problem states the test correctly identifies **98.2%** of people who have the disease (this is the \"Power\" or Sensitivity of the test).<BR>\n" );
document.write( "* The Type II error is the remaining percentage of people with the disease who were missed by the test.\r<BR>\n" );
document.write( "\n" );
document.write( "**Calculation:**<BR>\n" );
document.write( "$$P(\text{Type II Error}) = 100\% - 98.2\%$$<BR>\n" );
document.write( "$$P(\text{Type II Error}) = 1.8\% = 0.018$$\r<BR>\n" );
document.write( "\n" );
document.write( "> **Probability of a Type II error: 0.018** </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );