document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1201344>Question 1201344</A>: <I><SPAN STYLE=\"word-break: break-all;\"> State the null and alternative hypotheses, find the P  -  Value, find the critical value,<BR>\n" );
document.write( "should we reject Ho, or should we fail to reject Ho at 5% level of significance (why or why<BR>\n" );
document.write( "not?) in the following situations:\r<BR>\n" );
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document.write( "a. The test statistic of Z = 1.00 is obtained when testing the claim that p > 0.3\r<BR>\n" );
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document.write( "b. The test statistic of Z = -2.50 is obtained when testing the claim that P < 0.75\r<BR>\n" );
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document.write( "c. The test statistic of Z = - 1.94 is obtained when testing the claim that p = 0.375<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #835808 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=835808\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Part (a)\r<BR>\n" );
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document.write( "Null: p = 0.3<BR>\n" );
document.write( "Alternative: p > 0.3\r<BR>\n" );
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document.write( "Rule: The null ALWAYS will have the equal sign. <BR>\n" );
document.write( "This is to lock down the parameter we're trying to test.<BR>\n" );
document.write( "The alternative will have an inequality sign of some sort (eg: \"greater than\")\r<BR>\n" );
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document.write( "The claim is p > 0.3 which tells us the direction of the test. We're doing a right-tailed test. <BR>\n" );
document.write( "The rejection region is to the right of the critical value because of this right-tailed test.\r<BR>\n" );
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document.write( "At the 5% level of significance, the z critical value is roughly z = 1.645<BR>\n" );
document.write( "This is because P(Z > 1.645) = 0.05<BR>\n" );
document.write( "Approximately 5% of the area under the Z curve is to the right of 1.645.<BR>\n" );
document.write( "Use a table or calculator to determine this value.\r<BR>\n" );
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document.write( "Use a Z table such as this<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.ztable.net/\">https://www.ztable.net/</a><BR>\n" );
document.write( "or the table found in the back of your stats textbook\r<BR>\n" );
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document.write( "Use such a table to find that<BR>\n" );
document.write( "P(Z < 1.00) = 0.84134<BR>\n" );
document.write( "so,<BR>\n" );
document.write( "P(Z > 1.00) = 1 - P(Z < 1.00)<BR>\n" );
document.write( "P(Z > 1.00) = 1 - 0.84134<BR>\n" );
document.write( "P(Z > 1.00) = 0.15866<BR>\n" );
document.write( "This is the approximate p-value.\r<BR>\n" );
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document.write( "Rule: If the p-value is smaller than alpha, reject the null.\r<BR>\n" );
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document.write( "We have<BR>\n" );
document.write( "p-value = 0.15866<BR>\n" );
document.write( "alpha = 0.05<BR>\n" );
document.write( "Therefore, we fail to reject the null.\r<BR>\n" );
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document.write( "Take note how the test statistic z = 1.00 is not to the right of the critical value z = 1.645, so we are not in the rejection region, and have further evidence to fail to reject the null.\r<BR>\n" );
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document.write( "<font color=red>Summary:</font><BR>\n" );
document.write( "Null: p = 0.3<BR>\n" );
document.write( "Alternative: p > 0.3<BR>\n" );
document.write( "P-value: 0.15866  (approximate)<BR>\n" );
document.write( "Critical value: z = 1.645 (approximate)<BR>\n" );
document.write( "Decision: fail to reject the null (aka accept the null)\r<BR>\n" );
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document.write( "Part (b)\r<BR>\n" );
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document.write( "Null: p = 0.75<BR>\n" );
document.write( "Alternative: p < 0.75<BR>\n" );
document.write( "This is a left-tailed test\r<BR>\n" );
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document.write( "The critical value is now z = -1.645 because we're doing a left-tailed test<BR>\n" );
document.write( "P(Z < -1.645) = 0.05 approximately\r<BR>\n" );
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document.write( "Use a Z table to find that<BR>\n" );
document.write( "P(Z < -2.50) = 0.00621<BR>\n" );
document.write( "which is the approximate p-value.\r<BR>\n" );
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document.write( "The p-value is smaller than alpha = 0.05, so we will reject the null. Notice the test statistic is to the left of the critical value. The test statistic is in the rejection region.\r<BR>\n" );
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document.write( "<font color=red>Summary:</font><BR>\n" );
document.write( "Null: p = 0.75<BR>\n" );
document.write( "Alternative: p < 0.75<BR>\n" );
document.write( "P-value: 0.00621  (approximate)<BR>\n" );
document.write( "Critical value: z = -1.645 (approximate)<BR>\n" );
document.write( "Decision: Reject the null\r<BR>\n" );
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document.write( "Part (c)\r<BR>\n" );
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document.write( "Null: p = 0.375<BR>\n" );
document.write( "Alternative: p =/= 0.375<BR>\n" );
document.write( "The null will ALWAYS have the equal sign<BR>\n" );
document.write( "In this case, the alternative hypothesis has the not equal sign (this is a two tailed test because of that).\r<BR>\n" );
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document.write( "Use a calculator or table to find the z critical values are roughly z = -1.960 and z = 1.960; a two-tailed test requires two critical values.<BR>\n" );
document.write( "P(-1.960 < Z < 1.960) = 0.95 approximately<BR>\n" );
document.write( "which can be rephrased as<BR>\n" );
document.write( "P(Z < -1.960 or Z > 1.960) = 0.05 approximately\r<BR>\n" );
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document.write( "Rule: if the test statistic is between the critical values, then fail to reject the null. Otherwise, reject the null.\r<BR>\n" );
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document.write( "z = -1.94 is between -1.960 and 1.960<BR>\n" );
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document.write( "Therefore, we fail to reject the null.\r<BR>\n" );
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document.write( "Use a Z table to find that<BR>\n" );
document.write( "P(Z < -1.94) = 0.02619<BR>\n" );
document.write( "double this value because we're doing a two-tailed test<BR>\n" );
document.write( "2*0.02619 = 0.05238\r<BR>\n" );
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document.write( "The p-value is roughly 0.05238<BR>\n" );
document.write( "It is not smaller than alpha = 0.05, so this is further evidence we fail to reject the null.\r<BR>\n" );
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document.write( "<font color=red>Summary:</font><BR>\n" );
document.write( "Null: p = 0.375<BR>\n" );
document.write( "Alternative: p =/= 0.375<BR>\n" );
document.write( "P-value: 0.05238  (approximate)<BR>\n" );
document.write( "Critical values:  z = -1.960 and z = 1.960 (approximate)<BR>\n" );
document.write( "Decision: Fail to reject the null<BR>\n" );
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