document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1207903>Question 1207903</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Independent random samples of n1 = 18 and n2 = 13 observations were selected from two normal populations with equal variances.<BR>\n" );
document.write( "DATA:-<BR>\n" );
document.write( "____________________Population   (Ignore the lines - it represents space) <BR>\n" );
document.write( "____________________ 1_____2     (Ignore the lines - it represents space)<BR>\n" );
document.write( "Sample Size_________18____13     (Ignore the lines - it represents space)<BR>\n" );
document.write( "Sample Mean________34.6___32.1   (Ignore the lines - it represents space)<BR>\n" );
document.write( "Sample Variance_____4.5___5.9    (Ignore the lines - it represents space)\r<BR>\n" );
document.write( "\n" );
document.write( "(a) Find the rejection region for the test in part (a) for 𝛼 = 0.01. (If the test is one-tailed, enter NONE for the unused region. Round your answers to three decimal places.)<BR>\n" );
document.write( "t > _______<BR>\n" );
document.write( "t < _______\r<BR>\n" );
document.write( "\n" );
document.write( "(b) Find the value of the test statistic. (Round your answer to three decimal places.)<BR>\n" );
document.write( "t = \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "(c) Find the approximate p-value for the test.<BR>\n" );
document.write( "p-value < 0.010<BR>\n" );
document.write( "i) 0.010 < p-value < 0.020    <BR>\n" );
document.write( "ii) 0.020 < p-value < 0.050<BR>\n" );
document.write( "iii) 0.050 < p-value < 0.100<BR>\n" );
document.write( "iv) 0.100 < p-value < 0.200<BR>\n" );
document.write( "v) p-value < 0.200 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848015 by </B><A HREF=/tutors/aboutme.mpl?userid=asinus><B>asinus(45)</B></A><img src=\"/images/stars/stars-07.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=asinus><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=848015\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> To address this hypothesis testing problem, we'll break down the process step-by-step. Here are the steps for parts (a), (b), and (c):\r<BR>\n" );
document.write( "\n" );
document.write( "### (a) Find the Rejection Region for the Test\r<BR>\n" );
document.write( "\n" );
document.write( "In hypothesis testing, the rejection region is determined based on the significance level (α). Given α = 0.01, we need to find critical values for the t-distribution with the appropriate degrees of freedom.\r<BR>\n" );
document.write( "\n" );
document.write( "1. **Degrees of freedom (df)**: Since we have two samples, we can calculate the degrees of freedom using the formula:<BR>\n" );
document.write( "   \[<BR>\n" );
document.write( "   df = n_1 + n_2 - 2<BR>\n" );
document.write( "   \]<BR>\n" );
document.write( "   where \(n_1 = 18\) and \(n_2 = 13\). Therefore,<BR>\n" );
document.write( "   \[<BR>\n" );
document.write( "   df = 18 + 13 - 2 = 29<BR>\n" );
document.write( "   \]\r<BR>\n" );
document.write( "\n" );
document.write( "2. **Determine the critical t-value**: Since we're not told whether this is a one-tailed or two-tailed test, I'll assume it's a two-tailed test for the sake of this example. For α = 0.01 (which is split between the two tails), each tail would contain 0.005. You can find the critical t-value using a t-table or a calculator for \(df = 29\) and α/2 = 0.005.\r<BR>\n" );
document.write( "\n" );
document.write( "   Using a t-table or calculator, you would find:<BR>\n" );
document.write( "   - Critical t values for df = 29 at the 0.005 level.\r<BR>\n" );
document.write( "\n" );
document.write( "   For a two-tailed test:<BR>\n" );
document.write( "   - \(t < t_{\alpha/2}\) (negative critical value) and \(t > t_{(1-\alpha/2)}\) (positive critical value).<BR>\n" );
document.write( "   <BR>\n" );
document.write( "   Let's assume that the t-table shows critical values of:<BR>\n" );
document.write( "   \[<BR>\n" );
document.write( "   t_{\alpha/2} \approx \pm 2.756<BR>\n" );
document.write( "   \]<BR>\n" );
document.write( "   Thus, the rejection region will be:<BR>\n" );
document.write( "   \[<BR>\n" );
document.write( "   t < -2.756 \quad \text{and} \quad t > 2.756<BR>\n" );
document.write( "   \]\r<BR>\n" );
document.write( "\n" );
document.write( "3. **Final Result**:<BR>\n" );
document.write( "   - For the rejection region:<BR>\n" );
document.write( "     - **t > 2.756**<BR>\n" );
document.write( "     - **t < -2.756**  \r<BR>\n" );
document.write( "\n" );
document.write( "### (b) Find the Value of the Test Statistic\r<BR>\n" );
document.write( "\n" );
document.write( "To compute the t-statistic, we can use the following formula for two independent samples:<BR>\n" );
document.write( "\[<BR>\n" );
document.write( "t = \frac{\bar{X_1} - \bar{X_2}}{S_p\sqrt{\frac{1}{n_1} + \frac{1}{n_2}}}<BR>\n" );
document.write( "\]<BR>\n" );
document.write( "where:<BR>\n" );
document.write( "- \(\bar{X_1} = 34.6\) (mean of the first sample)<BR>\n" );
document.write( "- \(\bar{X_2} = 32.1\) (mean of the second sample)<BR>\n" );
document.write( "- \(S_p\) is the pooled standard deviation, calculated as:<BR>\n" );
document.write( "\[<BR>\n" );
document.write( "S_p = \sqrt{\frac{(n_1 - 1)S_1^2 + (n_2 - 1)S_2^2}{n_1 + n_2 - 2}}<BR>\n" );
document.write( "\]\r<BR>\n" );
document.write( "\n" );
document.write( "Given:<BR>\n" );
document.write( "- Sample variance \(S_1^2 = 4.5\)<BR>\n" );
document.write( "- Sample variance \(S_2^2 = 5.9\)\r<BR>\n" );
document.write( "\n" );
document.write( "1. Calculate \(S_p\):<BR>\n" );
document.write( "   \[<BR>\n" );
document.write( "   S_p = \sqrt{\frac{(18 - 1) \cdot 4.5 + (13 - 1) \cdot 5.9}{18 + 13 - 2}} = \sqrt{\frac{17 \cdot 4.5 + 12 \cdot 5.9}{29}} <BR>\n" );
document.write( "   \]<BR>\n" );
document.write( "   \[<BR>\n" );
document.write( "   = \sqrt{\frac{76.5 + 70.8}{29}} = \sqrt{\frac{147.3}{29}} \approx \sqrt{5.08} \approx 2.253<BR>\n" );
document.write( "   \]\r<BR>\n" );
document.write( "\n" );
document.write( "2. Now, substitute back into the t formula:<BR>\n" );
document.write( "\[<BR>\n" );
document.write( "t = \frac{34.6 - 32.1}{2.253\sqrt{\frac{1}{18} + \frac{1}{13}}}<BR>\n" );
document.write( "\]<BR>\n" );
document.write( "Calculate the individual components:<BR>\n" );
document.write( "\[<BR>\n" );
document.write( "\sqrt{\frac{1}{18} + \frac{1}{13}} = \sqrt{0.05556 + 0.07692} = \sqrt{0.13248} \approx 0.363<BR>\n" );
document.write( "\]\r<BR>\n" );
document.write( "\n" );
document.write( "And now find \(t\):<BR>\n" );
document.write( "\[<BR>\n" );
document.write( "t = \frac{2.5}{2.253 \times 0.363} \approx \frac{2.5}{0.817} \approx 3.059<BR>\n" );
document.write( "\]\r<BR>\n" );
document.write( "\n" );
document.write( "So:<BR>\n" );
document.write( "\[<BR>\n" );
document.write( "t \approx 3.059 \quad \text{(rounded to three decimal places)}<BR>\n" );
document.write( "\]\r<BR>\n" );
document.write( "\n" );
document.write( "### (c) Find the Approximate p-value for the Test\r<BR>\n" );
document.write( "\n" );
document.write( "To find the p-value associated with the test statistic \(t \approx 3.059\) and \(df = 29\):\r<BR>\n" );
document.write( "\n" );
document.write( "1. Because this is a one-tailed test, you would look up the tail probability of \(t = 3.059\) in the t-distribution table or use statistical software/calculator.\r<BR>\n" );
document.write( "\n" );
document.write( "2. Generally, for values beyond \(t > 2.756\), we know that the p-value will be less than alpha = 0.01. Thus, we can check ranges. For \(t \approx 3.059\), it is likely to be in the range of < 0.01.\r<BR>\n" );
document.write( "\n" );
document.write( "After checking the statistical tables or calculations, we find that:<BR>\n" );
document.write( "\[<BR>\n" );
document.write( "\text{p-value} < 0.010<BR>\n" );
document.write( "\]\r<BR>\n" );
document.write( "\n" );
document.write( "### Conclusion\r<BR>\n" );
document.write( "\n" );
document.write( "So, your answers would be:<BR>\n" );
document.write( "- (a) \( t > 2.756 \); \( t < -2.756 \)<BR>\n" );
document.write( "- (b) \( t = 3.059 \)<BR>\n" );
document.write( "- (c) p-value < 0.010\r<BR>\n" );
document.write( "\n" );
document.write( "Feel free to ask if you have any further questions or need clarification on any of the steps! </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );