Hypothesis Testing: Mean of a Large Population

Given Data

1. Sample Mean ((\bar{X})): 3.4 cm
2. Population Mean ((\mu)): 3.25 cm
3. Standard Deviation ((\sigma)): 2.61 cm
4. Sample Size ((n)): 900
5. Level of Significance ((\alpha)): 5% (0.05)

Hypotheses

• Null Hypothesis ((H_0)): (\mu = 3.25) cm
• Alternate Hypothesis ((H_1)): (\mu \neq 3.25) cm

Test Statistics Formula

[ Z = \frac{\bar{X} - \mu}{\frac{\sigma}{\sqrt{n}}} ]

Calculations

1. Sample Mean: (\bar{X} = 3.4) cm
2. Population Mean: (\mu = 3.25) cm
3. Difference ((\bar{X} - \mu)): [ 3.4 - 3.25 = 0.15 ]
4. **Standard Error ((SE)): [ \frac{\sigma}{\sqrt{n}} = \frac{2.61}{\sqrt{900}} = \frac{2.61}{30} = 0.087 ]
5. Test Statistic (Z-value): [ Z = \frac{0.15}{0.087} = 1.72 ]

Comparison with Critical Value

• Critical Value at 5% Significance Level (Two-tailed test): 1.96
• Calculated Z-value: 1.72

Since (|Z_{calculated}| < Z_{critical}), we fail to reject the null hypothesis.

Conclusion

• At the 5% significance level, the sample mean of 3.4 cm is not significantly different from the population mean of 3.25 cm.

95% Confidence Interval for the Sample Mean

• Formula: [ CI = \bar{X} \pm Z_{\alpha/2} \times \frac{\sigma}{\sqrt{n}} ] where: ( Z_{\alpha/2} = 1.96 )
• Calculation:
  1. ( Margin \ of \ Error = 1.96 \times 0.087 = 0.171 )
  2. [ CI = 3.4 \pm 0.171 ]
  3. [ CI = (3.229, 3.571) ]
• Interpretation: We are 95% confident that the true population mean lies between 3.229 cm and 3.571 cm.