document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1193764>Question 1193764</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A data set about speed dating includes like\" ratings of male dates made by the female dates. The summary statistics are n = 191 , x = 5.83 , s = 1.98 . Use a 0.05 significance level to test the claim that the population mean of such ratings is less than 6.00. Assume that a simple random sample has been selected. Identify the null and alternative hypotheses test statistic, P-value, and state the final conclusion that addresses the original claim.\r<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 #825801 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=825801\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "mu = population mean of the rating\r<BR>\n" );
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document.write( "Claim: \"The population mean of such ratings is less than 6.00\"<BR>\n" );
document.write( "Translation into symbols: mu < 6.00<BR>\n" );
document.write( "The goal of this entire solution is to test if this claim is true or not.\r<BR>\n" );
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document.write( "Hypotheses:<BR>\n" );
document.write( "H0: mu = 6.00<BR>\n" );
document.write( "H1: mu < 6.00<BR>\n" );
document.write( "Claim is in the alternative hypothesis H1<BR>\n" );
document.write( "We have a left-tailed test because of the inequality sign here.<BR>\n" );
document.write( "Some stats textbooks would state the null as mu ≥ 6.00, but I'll stick with mu = 6.00 so that we focus on exactly one mu value only. The null must be centered around one single value. \r<BR>\n" );
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document.write( "Given info:<BR>\n" );
document.write( "n = 191 = sample size<BR>\n" );
document.write( "xbar = 5.83 = sample mean<BR>\n" );
document.write( "s = 1.98 = sample standard deviation\r<BR>\n" );
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document.write( "Test Statistic:<BR>\n" );
document.write( "z = (xbar - mu)/( s/sqrt(n) )<BR>\n" );
document.write( "z = (5.83 - 6.00)/( 1.98/sqrt(1.91) )<BR>\n" );
document.write( "z = -0.11865892643357<BR>\n" );
document.write( "z = -0.12<BR>\n" );
document.write( "Despite sigma being unknown, we can use z here because n > 30 is true. The student T distribution is very similar to the standard normal Z distribution for n > 30.\r<BR>\n" );
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document.write( "Use a calculator like this<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://onlinestatbook.com/2/calculators/normal_dist.html\">https://onlinestatbook.com/2/calculators/normal_dist.html</a><BR>\n" );
document.write( "to find that P(Z < -0.12) = 0.4522 approximately<BR>\n" );
document.write( "If you have your TI calculator with you and wish to use it, then hit the key labeled \"2ND\" and hit the \"VARS\" button to bring up the stats menu. The function you want is called normalcdf. <BR>\n" );
document.write( "You can type in something like normalcdf(-99, -0.12) and you should get some value close to 0.4522\r<BR>\n" );
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document.write( "Alternatively, you can use a Z table like this<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.ztable.net/\">https://www.ztable.net/</a><BR>\n" );
document.write( "to find that P(Z < -0.12) = 0.45224 approximately.<BR>\n" );
document.write( "Look at the row that starts with -0.1 and the column that has 0.02 up top. The two combine to have 0.45224 \r<BR>\n" );
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document.write( "Note: We shade to the left of z = -0.12 directly because we're doing a left-tailed test.\r<BR>\n" );
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document.write( "The p-value is roughly 0.4522<BR>\n" );
document.write( "It is the probability of selecting a z score of z = -0.12 or smaller. It is also the probability of getting xbar = 5.83 or smaller.\r<BR>\n" );
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document.write( "Comparing the p-value (0.4522) and the alpha value (0.05), we see the p-value is clearly much larger. <BR>\n" );
document.write( "Therefore, we fail to reject the null. We only reject the null when the p-value is smaller than alpha.<BR>\n" );
document.write( "It is said we \"accept\" the null. I put that in quotes because we don't really accept it, rather we just don't have enough data to overturn it.<BR>\n" );
document.write( "Side note: The alpha level is the probability of a Type I error.\r<BR>\n" );
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document.write( "Since we failed to reject the null, we have no choice but to \"accept\" that mu = 6.00 is the case.<BR>\n" );
document.write( "To address the original claim, we have shown it to be false because mu is not less than 6.00<BR>\n" );
document.write( "Recall the claim was made in the alternative hypothesis. <BR>\n" );
document.write( "We don't have enough statistically significant evidence to overturn the null and side with the alternative.\r<BR>\n" );
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document.write( "Yes the xbar = 5.83 is smaller than 6.00, but this appears to be due to random chance.<BR>\n" );
document.write( "If you were to do another random sample of 191, then you might get some xbar equal to 6.00 or larger.\r<BR>\n" );
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document.write( "Summary:\r<BR>\n" );
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document.write( "Null hypothesis: H0: mu = 6.00<BR>\n" );
document.write( "Alternative hypothesis: H1: mu < 6.00\r<BR>\n" );
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document.write( "Test Statistic: z = -0.12<BR>\n" );
document.write( "P-value = 0.4522\r<BR>\n" );
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document.write( "Decision: Fail to reject the null (i.e. \"accept\" the null)<BR>\n" );
document.write( "Conclusion: The population mean of the ratings is not less than 6.00 (thereby the claim made at the top is false)<BR>\n" );
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