document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1166031>Question 1166031</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A sales clerk in the departmental store claims that 65% of the shoppers entering the store leave without making a purchase. A random sample of 60 shoppers showed that 40 of them left without buying anything. Are these sample results consistent with the claim of the sales clerk at 1% level of significance?                                                         </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #852871 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2189)</B></A><img src=\"/images/stars/stars-13.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=852871\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> This problem requires a **one-sample proportion Z-test** to determine if the sample results are consistent with the sales clerk's claim. \r<BR>\n" );
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document.write( "## 📝 Hypothesis Test for Population Proportion\r<BR>\n" );
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document.write( "### 1. Define Hypotheses\r<BR>\n" );
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document.write( "The sales clerk claims that 65% of shoppers leave without purchasing, so the null hypothesis ($H_0$) is that the true population proportion ($p$) is $0.65$. The alternative hypothesis ($H_1$) tests if the sample is *inconsistent* with this claim, suggesting the proportion is different.\r<BR>\n" );
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document.write( "* **Null Hypothesis ($H_0$):** $p = 0.65$ (The true proportion of shoppers leaving without a purchase is 65%).<BR>\n" );
document.write( "* **Alternative Hypothesis ($H_1$):** $p \neq 0.65$ (The true proportion is different from 65%).\r<BR>\n" );
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document.write( "This is a **two-tailed test**.\r<BR>\n" );
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document.write( "### 2. Determine the Level of Significance and Critical Values\r<BR>\n" );
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document.write( "* **Level of Significance ($\alpha$):** $1\%$ or $0.01$.<BR>\n" );
document.write( "* Since this is a two-tailed test, the significance is split between the two tails ($\alpha/2 = 0.01/2 = 0.005$).<BR>\n" );
document.write( "* The **critical Z-values** corresponding to an area of $0.005$ in each tail are found from the standard normal distribution table:<BR>\n" );
document.write( "    $$Z_{\text{critical}} = \mathbf{\pm 2.58}$$\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### 3. Calculate Test Statistic\r<BR>\n" );
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document.write( "We use the sample data to calculate the test statistic ($Z_{\text{calc}}$).\r<BR>\n" );
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document.write( "#### A. Sample Proportion ($\hat{p}$)<BR>\n" );
document.write( "* Sample size ($n$) = 60<BR>\n" );
document.write( "* Number of shoppers leaving without purchase ($x$) = 40<BR>\n" );
document.write( "* Sample proportion ($\hat{p}$) is the observed proportion:<BR>\n" );
document.write( "    $$\hat{p} = \frac{x}{n} = \frac{40}{60} \approx \mathbf{0.6667}$$\r<BR>\n" );
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document.write( "#### B. Standard Error ($SE$)<BR>\n" );
document.write( "We use the claimed population proportion ($p_0 = 0.65$) to calculate the standard error for the test statistic:<BR>\n" );
document.write( "$$SE = \sqrt{\frac{p_0 (1 - p_0)}{n}} = \sqrt{\frac{0.65 (1 - 0.65)}{60}}$$<BR>\n" );
document.write( "$$SE = \sqrt{\frac{0.65 \times 0.35}{60}} = \sqrt{\frac{0.2275}{60}} \approx \sqrt{0.003792} \approx \mathbf{0.06158}$$\r<BR>\n" );
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document.write( "#### C. Calculated Z-Score ($Z_{\text{calc}}$)<BR>\n" );
document.write( "$$Z_{\text{calc}} = \frac{\hat{p} - p_0}{SE} = \frac{0.6667 - 0.65}{0.06158}$$<BR>\n" );
document.write( "$$Z_{\text{calc}} = \frac{0.0167}{0.06158} \approx \mathbf{0.27}$$\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### 4. Decision and Conclusion\r<BR>\n" );
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document.write( "* **Decision Rule:** Reject $H_0$ if $|Z_{\text{calc}}| > |Z_{\text{critical}}|$.<BR>\n" );
document.write( "* **Comparison:**<BR>\n" );
document.write( "    $$|0.27| \text{ is not greater than } |2.58|$$<BR>\n" );
document.write( "    $0.27 < 2.58$\r<BR>\n" );
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document.write( "The calculated test statistic ($Z_{\text{calc}} = 0.27$) falls within the non-rejection region (the area between -2.58 and 2.58).\r<BR>\n" );
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document.write( "**Conclusion:**\r<BR>\n" );
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document.write( "Since the test statistic is not in the critical region, we **fail to reject the null hypothesis ($H_0$)**.\r<BR>\n" );
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document.write( "The sample results ($\hat{p} \approx 66.67\%$) are **consistent with the claim** of the sales clerk ($p = 65\%$) at the $1\%$ level of significance. The small difference observed is attributed to **random sampling variation**. </SPAN>\n" );
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