document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1185821>Question 1185821</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A study is done by a community group in two neighboring colleges to determine which graduate students with more maths classes. College A samples 11 graduates. Their average is 4 maths classes with a standard deviation of 1.5 math classes. College B samples 9 grad. Their average is 3.5 maths classes with a standard deviation of 1 maths class. The community group believes that a student who graduate from college A has taken more maths classes, on the average. Both population have a normal distribution. Test at a 1% significance level.\r<BR>\n" );
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document.write( "(a) is this a test two means or two proportions?<BR>\n" );
document.write( "(b) Are the populations' standard deviation known or unknown?<BR>\n" );
document.write( "(c) Which distribution do you use to perform the test?<BR>\n" );
document.write( "(d) What is the random variables?<BR>\n" );
document.write( "(e) What is the null and alternative hypothesis?<BR>\n" );
document.write( "(f) is this test right, left, or two tailed<BR>\n" );
document.write( "(g) What is the p-value?<BR>\n" );
document.write( "(h) Do you reject or not reject the null hypothesis? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849610 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1987)</B></A><img src=\"/images/stars/stars-12.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=849610\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to address each part of the hypothesis test:\r<BR>\n" );
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document.write( "**(a) Two Means or Two Proportions?**\r<BR>\n" );
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document.write( "This is a test of *two means* because we are comparing the average number of math classes taken by graduates from two different colleges.\r<BR>\n" );
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document.write( "**(b) Known or Unknown Population Standard Deviations?**\r<BR>\n" );
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document.write( "The population standard deviations are *unknown*. We are given the *sample* standard deviations.\r<BR>\n" );
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document.write( "**(c) Which Distribution to Use?**\r<BR>\n" );
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document.write( "Since the population standard deviations are unknown and the sample sizes are small (n1 = 11, n2 = 9), we use the *t-distribution*.  Because the sample sizes are small, and the population standard deviations are unknown, we must assume that the population variances are equal.\r<BR>\n" );
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document.write( "**(d) What is the Random Variable?**\r<BR>\n" );
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document.write( "The random variable is the *difference* between the two sample means:  x̄1 - x̄2, where x̄1 is the mean number of math classes for College A graduates and x̄2 is the mean number of math classes for College B graduates.\r<BR>\n" );
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document.write( "**(e) Null and Alternative Hypotheses:**\r<BR>\n" );
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document.write( "*   **Null Hypothesis (H0):** The average number of math classes taken by graduates from College A is equal to or less than the average number of math classes taken by graduates from College B.  μ1 ≤ μ2\r<BR>\n" );
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document.write( "*   **Alternative Hypothesis (H1):** The average number of math classes taken by graduates from College A is greater than the average number of math classes taken by graduates from College B. μ1 > μ2  (This is what the community group believes.)\r<BR>\n" );
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document.write( "**(f) Right, Left, or Two-Tailed Test?**\r<BR>\n" );
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document.write( "This is a *right-tailed* test because the alternative hypothesis states that the mean for College A is *greater* than the mean for College B.\r<BR>\n" );
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document.write( "**(g) What is the p-value?**\r<BR>\n" );
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document.write( "1.  **Calculate the test statistic:**<BR>\n" );
document.write( "    Because we are assuming the population variances are equal, we use the following formula for the t-statistic:<BR>\n" );
document.write( "    ```<BR>\n" );
document.write( "    t = (x̄1 - x̄2) / sqrt(s_p^2 * (1/n1 + 1/n2))<BR>\n" );
document.write( "    ```<BR>\n" );
document.write( "    Where s_p is the pooled standard deviation, calculated as:<BR>\n" );
document.write( "    ```<BR>\n" );
document.write( "    s_p = sqrt(((n1-1)*s1^2 + (n2-1)*s2^2) / (n1 + n2 - 2))<BR>\n" );
document.write( "    ```<BR>\n" );
document.write( "    Plugging in the given values, we get s_p = sqrt(((10)*1.5^2 + (8)*1^2) / 18) = 1.32<BR>\n" );
document.write( "    Then, the t statistic is calculated as:<BR>\n" );
document.write( "    ```<BR>\n" );
document.write( "    t = (4 - 3.5) / sqrt(1.32^2 * (1/11 + 1/9)) = 0.5/(1.32*0.447) = 0.84<BR>\n" );
document.write( "    ```\r<BR>\n" );
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document.write( "2.  **Degrees of Freedom:** df = n1 + n2 - 2 = 11 + 9 - 2 = 18\r<BR>\n" );
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document.write( "3.  **Find the p-value:** Using a t-table or calculator with df = 18 and t = 0.84, we find the p-value.  Since it is a right-tailed test, the p-value is the area to the *right* of the calculated t-statistic.  The p-value is approximately 0.208.\r<BR>\n" );
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document.write( "**(h) Reject or Not Reject the Null Hypothesis?**\r<BR>\n" );
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document.write( "*   **Significance Level:** α = 0.01\r<BR>\n" );
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document.write( "*   **Decision:** Since the p-value (0.208) is *greater* than the significance level (0.01), we *fail to reject* the null hypothesis.\r<BR>\n" );
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document.write( "**(Conclusion):**\r<BR>\n" );
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document.write( "At a 1% significance level, there is *not* sufficient evidence to conclude that graduates from College A take more math classes on average than graduates from College B.<BR>\n" );
document.write( " </SPAN>\n" );
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