Statistics Summary Notes

Study Time for UNF Students

  • Point Estimate for Average Study Time

    • Notation: ( \bar{x} )
    • Value: 9.3 hours

  • 90% Confidence Interval Calculation

    1. Sample Size (n): 70
    2. Sample Mean (( \bar{x} )): 9.3 hours
    3. Standard Deviation (s): 2.7 hours
    4. Find the critical value (z) for 90% confidence:
    • Using z-table, ( z \approx 1.645 )
    1. Calculate the Standard Error (SE):
    • ( SE = \frac{s}{\sqrt{n}} = \frac{2.7}{\sqrt{70}} \approx 0.323 )
    1. Margin of Error (ME):
    • ( ME = z \times SE = 1.645 \times 0.323 \approx 0.531 )
    1. Confidence Interval:
    • Lower limit: ( \bar{x} - ME = 9.3 - 0.531 = 8.769 )
    • Upper limit: ( \bar{x} + ME = 9.3 + 0.531 = 9.831 )
    • Final Confidence Interval: ( (8.769, 9.831) ) hours

  • Plausibility of Average Study Time being 5.8 hours

    • Since 5.8 hours is outside the confidence interval of (8.769, 9.831), it is not plausible that the average study time is 5.8 hours.

Proportion of American Households with Dogs

  • Point Estimate for Proportion of Households with Dogs

    • Notation: ( \hat{p} )
    • Value: ( \hat{p} = \frac{184}{400} = 0.46 )

  • 95% Confidence Interval Calculation

    1. Sample Size (n): 400
    2. Sample Proportion (( \hat{p} )): 0.46
    3. Standard Error (SE):
    • ( SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} = \sqrt{\frac{0.46 \times (1 - 0.46)}{400}} \approx 0.025 )
    1. Critical value (z) for 95% confidence: ( z \approx 1.96 )
    2. Margin of Error (ME):
    • ( ME = z \times SE = 1.96 \times 0.025 \approx 0.049 )
    1. Confidence Interval:
    • Lower limit: ( \hat{p} - ME = 0.46 - 0.049 = 0.411 )
    • Upper limit: ( \hat{p} + ME = 0.46 + 0.049 = 0.509 )
    • Final Confidence Interval: ( (0.411, 0.509) )

  • Margin of Error Calculation

    • Value: 0.049

  • Plausibility of Proportion being 0.45

    • 0.45 falls within the confidence interval of (0.411, 0.509), so it is plausible that the proportion of all American households that have a dog is 0.45.

Tensile Strength of Thread

  • Point Estimate for Population Mean Tensile Strength

    • Notation: ( \bar{x} )
    • Calculate the mean of the sample:
    • Sample data: 92.7, 91.1, 89.9, 93.3, 94.1, 91.4, 89.6, 91.8
    • ( \bar{x} = \frac{92.7 + 91.1 + 89.9 + 93.3 + 94.1 + 91.4 + 89.6 + 91.8}{8} = 91.63 ) kg

  • 99% Confidence Interval Calculation

    1. Sample Size (n): 8
    2. Sample Mean (( \bar{x} )): 91.63 kg
    3. Known Standard Deviation (( \sigma )): 3.12 kg
    4. Critical value (z) for 99% confidence: ( z \approx 2.576 )
    5. Standard Error (SE):
    • ( SE = \frac{\sigma}{\sqrt{n}} = \frac{3.12}{\sqrt{8}} \approx 1.101 )
    1. Margin of Error (ME):
    • ( ME = z \times SE = 2.576 \times 1.101 \approx 2.834 )
    1. Confidence Interval:
    • Lower limit: ( \bar{x} - ME = 91.63 - 2.834 = 88.796 )
    • Upper limit: ( \bar{x} + ME = 91.63 + 2.834 = 94.464 )
    • Final Confidence Interval: ( (88.796, 94.464) ) kg

  • Plausibility of Population Mean Tensile Strength being 91.6 kg

    • 91.6 kg falls within the confidence interval of (88.796, 94.464), so it is plausible that the population mean tensile strength is 91.6 kg.