document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1199646>Question 1199646</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A psycho-linguist has been studying the frequency of text-speech (e.g., R U Ok?, LOL ) has been progressing into EMAIL correspondence.  He collects a sample of 121 EMAIL messages and computes the mean (average) number of examples of text-speech per message.  He also has a huge library of archived EMAIL messages from the year 2001.  From these he is able to calculate the population average and standard deviation.  Compute a inferential statistic to evaluate the hypothesis that the use of text-speech has increased in the years since 2001.  Use an alpha level that would insure a maximum of a %5 risk of rejecting a true null hypothesis. \r<BR>\n" );
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document.write( "These are the obtained values:\r<BR>\n" );
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document.write( "Population Mean from 2001:                          1.6      \r<BR>\n" );
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document.write( "Population Standard deviation from 2001:    1.25    \r<BR>\n" );
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document.write( "Sample mean from this year:                           1.98     \r<BR>\n" );
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document.write( "Calculate and properly report your calculated statistic, and interpret your conclusion in one or two sentences. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848168 by </B><A HREF=/tutors/aboutme.mpl?userid=textot><B>textot(100)</B></A><img src=\"/images/stars/stars-08.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=textot><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=848168\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **1. Set up Hypotheses**\r<BR>\n" );
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document.write( "* **Null Hypothesis (H0):** μ = 1.6 (The mean number of text-speech instances in current emails is the same as in 2001)<BR>\n" );
document.write( "* **Alternative Hypothesis (H1):** μ > 1.6 (The mean number of text-speech instances in current emails is greater than in 2001)\r<BR>\n" );
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document.write( "**2. Calculate the Test Statistic (z-score)**\r<BR>\n" );
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document.write( "* Since we know the population standard deviation (σ = 1.25), we can use the z-test:<BR>\n" );
document.write( "   * z = (x̄ - μ) / (σ / √n) <BR>\n" );
document.write( "   * where:<BR>\n" );
document.write( "      * x̄ = Sample mean (1.98)<BR>\n" );
document.write( "      * μ = Population mean (1.6)<BR>\n" );
document.write( "      * σ = Population standard deviation (1.25)<BR>\n" );
document.write( "      * n = Sample size (121)\r<BR>\n" );
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document.write( "   * z = (1.98 - 1.6) / (1.25 / √121)<BR>\n" );
document.write( "   * z = 0.38 / (1.25 / 11)<BR>\n" );
document.write( "   * z = 0.38 / 0.1136 <BR>\n" );
document.write( "   * z ≈ 3.34\r<BR>\n" );
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document.write( "**3. Determine Critical Value**\r<BR>\n" );
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document.write( "* **Significance Level (α) = 0.05**<BR>\n" );
document.write( "* This is a one-tailed test (since H1 is μ > 1.6).<BR>\n" );
document.write( "* Find the critical z-value for α = 0.05 in a standard normal distribution table.<BR>\n" );
document.write( "* The critical z-value is approximately 1.645.\r<BR>\n" );
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document.write( "**4. Make a Decision**\r<BR>\n" );
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document.write( "* **Calculated z-score (3.34) is greater than the critical value (1.645).**<BR>\n" );
document.write( "* **Therefore, we reject the null hypothesis.**\r<BR>\n" );
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document.write( "**5. Conclusion**\r<BR>\n" );
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document.write( "* At the 0.05 significance level, there is sufficient evidence to conclude that the mean number of text-speech instances in current emails is significantly higher than the mean number of text-speech instances in emails from 2001.\r<BR>\n" );
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document.write( "**In summary:**\r<BR>\n" );
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document.write( "* The calculated z-score is 3.34.<BR>\n" );
document.write( "* We reject the null hypothesis at the 0.05 significance level. <BR>\n" );
document.write( "* This suggests that the use of text-speech in emails has significantly increased since 2001.<BR>\n" );
document.write( " </SPAN>\n" );
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