document.write( "Question 1198869: Calculate the standard error. May normality be assumed? (Round your answers to 4 decimal places.)\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "n = 27, m = 0.24
\n" ); document.write( "(b) n = 56, m = 0.43
\n" ); document.write( "(c)n = 123, m = 0.47
\n" ); document.write( "(d)n = 578, m = 0.004\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "
Algebra.Com's Answer #848225 by textot(100)\"\" \"About 
You can put this solution on YOUR website!
**Formula for Standard Error of the Proportion**\r
\n" ); document.write( "\n" ); document.write( "The standard error of the proportion (SE) is given by:\r
\n" ); document.write( "\n" ); document.write( "SE = √[p * (1 - p) / n] \r
\n" ); document.write( "\n" ); document.write( "where:\r
\n" ); document.write( "\n" ); document.write( "* p is the sample proportion
\n" ); document.write( "* n is the sample size\r
\n" ); document.write( "\n" ); document.write( "**Normality Assumption**\r
\n" ); document.write( "\n" ); document.write( "We can generally assume normality of the sampling distribution of the proportion if the following conditions are met:\r
\n" ); document.write( "\n" ); document.write( "* **n * p ≥ 10**
\n" ); document.write( "* **n * (1 - p) ≥ 10**\r
\n" ); document.write( "\n" ); document.write( "**Now let's calculate the standard error for each case and check for normality:**\r
\n" ); document.write( "\n" ); document.write( "**a) n = 27, m = 0.24**\r
\n" ); document.write( "\n" ); document.write( "* SE = √[0.24 * (1 - 0.24) / 27] = √[0.24 * 0.76 / 27] ≈ 0.0758
\n" ); document.write( "* n * p = 27 * 0.24 = 6.48
\n" ); document.write( "* n * (1 - p) = 27 * 0.76 = 20.52
\n" ); document.write( "* **Normality may not be assumed** because n * p < 10.\r
\n" ); document.write( "\n" ); document.write( "**b) n = 56, m = 0.43**\r
\n" ); document.write( "\n" ); document.write( "* SE = √[0.43 * (1 - 0.43) / 56] = √[0.43 * 0.57 / 56] ≈ 0.0658
\n" ); document.write( "* n * p = 56 * 0.43 = 24.08
\n" ); document.write( "* n * (1 - p) = 56 * 0.57 = 31.92
\n" ); document.write( "* **Normality can be assumed** as both n * p and n * (1 - p) are greater than 10.\r
\n" ); document.write( "\n" ); document.write( "**c) n = 123, m = 0.47**\r
\n" ); document.write( "\n" ); document.write( "* SE = √[0.47 * (1 - 0.47) / 123] = √[0.47 * 0.53 / 123] ≈ 0.0448
\n" ); document.write( "* n * p = 123 * 0.47 = 57.81
\n" ); document.write( "* n * (1 - p) = 123 * 0.53 = 65.19
\n" ); document.write( "* **Normality can be assumed** as both n * p and n * (1 - p) are greater than 10.\r
\n" ); document.write( "\n" ); document.write( "**d) n = 578, m = 0.004**\r
\n" ); document.write( "\n" ); document.write( "* SE = √[0.004 * (1 - 0.004) / 578] = √[0.004 * 0.996 / 578] ≈ 0.0026
\n" ); document.write( "* n * p = 578 * 0.004 = 2.312
\n" ); document.write( "* n * (1 - p) = 578 * 0.996 = 575.688
\n" ); document.write( "* **Normality may not be assumed** because n * p < 10.\r
\n" ); document.write( "\n" ); document.write( "**Summary**\r
\n" ); document.write( "\n" ); document.write( "| Sample Size (n) | Sample Proportion (m) | Standard Error (SE) | Normality Assumed? |
\n" ); document.write( "|---|---|---|---|
\n" ); document.write( "| 27 | 0.24 | 0.0758 | No |
\n" ); document.write( "| 56 | 0.43 | 0.0658 | Yes |
\n" ); document.write( "| 123 | 0.47 | 0.0448 | Yes |
\n" ); document.write( "| 578 | 0.004 | 0.0026 | No |\r
\n" ); document.write( "\n" ); document.write( "I hope this helps! Let me know if you have any other questions.
\n" ); document.write( " \n" ); document.write( "
\n" );