document.write( "Question 1184225: Using the 955 confidence level scores 97,93, 95, 97, 97,94, 92, 92,98,92, 93,94, 95, 96, 96, 96 you would expect the lower end of the confidence interval to be approximately what value? \n" ); document.write( "
Algebra.Com's Answer #849867 by CPhill(1959)\"\" \"About 
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Here's how to calculate the lower end of the 95% confidence interval for the given scores:\r
\n" ); document.write( "\n" ); document.write( "1. **Calculate the Sample Mean (x̄):**\r
\n" ); document.write( "\n" ); document.write( "Sum of scores = 97 + 93 + 95 + 97 + 97 + 94 + 92 + 92 + 98 + 92 + 93 + 94 + 95 + 96 + 96 + 96 = 1520
\n" ); document.write( "Number of scores (n) = 16
\n" ); document.write( "x̄ = 1520 / 16 = 95\r
\n" ); document.write( "\n" ); document.write( "2. **Calculate the Sample Standard Deviation (s):**\r
\n" ); document.write( "\n" ); document.write( "* Find the squared difference of each score from the mean: (97-95)²=4, (93-95)²=4, (95-95)²=0, (97-95)²=4, (97-95)²=4, (94-95)²=1, (92-95)²=9, (92-95)²=9, (98-95)²=9, (92-95)²=9, (93-95)²=4, (94-95)²=1, (95-95)²=0, (96-95)²=1, (96-95)²=1, (96-95)²=1
\n" ); document.write( "* Sum of squared differences = 4 + 4 + 0 + 4 + 4 + 1 + 9 + 9 + 9 + 9 + 4 + 1 + 0 + 1 + 1 + 1 = 60
\n" ); document.write( "* Variance (s²) = 60 / (16-1) = 60 / 15 = 4
\n" ); document.write( "* s = √4 = 2\r
\n" ); document.write( "\n" ); document.write( "3. **Find the t-value:**\r
\n" ); document.write( "\n" ); document.write( "Since the sample size is small (n < 30), we use the t-distribution. For a 95% confidence level and 15 degrees of freedom (n-1 = 16-1 = 15), the t-value is approximately 2.131. You can find this using a t-table or a calculator with statistical functions.\r
\n" ); document.write( "\n" ); document.write( "4. **Calculate the Margin of Error:**\r
\n" ); document.write( "\n" ); document.write( "Margin of Error = t * (s / √n)
\n" ); document.write( "Margin of Error = 2.131 * (2 / √16)
\n" ); document.write( "Margin of Error = 2.131 * (2 / 4)
\n" ); document.write( "Margin of Error = 2.131 * 0.5
\n" ); document.write( "Margin of Error ≈ 1.07\r
\n" ); document.write( "\n" ); document.write( "5. **Calculate the Confidence Interval:**\r
\n" ); document.write( "\n" ); document.write( "* Upper limit = x̄ + Margin of Error = 95 + 1.07 ≈ 96.07
\n" ); document.write( "* Lower limit = x̄ - Margin of Error = 95 - 1.07 ≈ 93.93\r
\n" ); document.write( "\n" ); document.write( "Therefore, the lower end of the 95% confidence interval is approximately 93.93.
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