1.1 1.2

Hospital Study on Girl Babies

1. Introduction to the Study

  • Focus: Proportions and confidence intervals for proportions in a hospital study regarding the birth of girl babies.

2. Data Given

  • Total babies born (n): 300

  • Girls born (x): 240

3. Best Point Estimate of the Population Proportion

3.1 Definition

  • The best point estimate for a population proportion (p) is the sample proportion ($\hat{p}$).

3.2 Calculation Formula

  • Formula: p=xnp = \frac{x}{n} where:

    • (x): Number of favorable outcomes (girls born)

    • (n): Total number of trials (total babies born)

3.3 Point Estimate Calculation

  • Substituting values:
    <br>p^=240300=0.8<br><br>\hat{p} = \frac{240}{300} = 0.8<br>

  • Result:

    • The best point estimate of the population proportion of girls born using this method is 0.8 or 80%.

4. 95% Confidence Interval Estimate of the Proportion

4.1 Procedure Overview

  • To construct a confidence interval (CI) for a population proportion, the following formula is used:
    <br>CI=p^±Z⋅p(1−p)n<br><br>CI = \hat{p} \pm Z \cdot \sqrt{\frac{p(1-p)}{n}}<br>

4.2 Variables Explained

  • $\hat{p}$: Sample proportion.

  • $Z$: Z-score corresponding to the desired confidence level (for 95% confidence, $Z \approx 1.96$).

  • $n$: Sample size.

4.3 Steps to Calculate the Confidence Interval
Step 1: Identify $Z_{\alpha/2}$

  • For a 95% confidence level:

    • Total alpha ($\alpha$): 0.05

    • $\alpha/2 = 0.025$

    • Z-score for this value from standard normal distribution is: 1.96.

Step 2: Calculate the Standard Error

  • The formula for Standard Error is:
    <br>SE=p(1−p)n<br><br>SE = \sqrt{\frac{p(1-p)}{n}}<br>

  • For this study:

    • p=0.8p = 0.8

    • Calculation:
      <br>SE=0.8(1−0.8)300=0.8×0.2300<br><br>SE = \sqrt{\frac{0.8(1-0.8)}{300}} = \sqrt{\frac{0.8 \times 0.2}{300}} <br>

    • Further calculation:
      <br>SE=0.16300<br><br>SE = \sqrt{\frac{0.16}{300}} <br>

    • Resulting in:
      <br>SE=0.0005333<br>ightarrow0.02309<br><br>SE = 0.0005333 <br>ightarrow 0.02309<br>

Step 3: Calculate the Margin of Error (E)

  • Margin of Error formula:
    <br>E=Z×SE<br><br>E = Z \times SE<br>

  • Substituting the values:

    • E=1.96×0.02309≈0.04526E = 1.96 \times 0.02309 \approx 0.04526

Step 4: Construct the Confidence Interval

  • The confidence interval is calculated as:
    <br>CI=p^±E<br><br>CI = \hat{p} \pm E<br>

  • Substituting in the values:

    • CI=0.8±0.04526CI = 0.8 \pm 0.04526

  • Calculating the bounds:

    • Lower Bound: 0.8−0.04526=0.754740.8 - 0.04526 = 0.75474

    • Upper Bound: 0.8+0.04526=0.845260.8 + 0.04526 = 0.84526

4.4 Final Result

  • The 95% confidence interval for the proportion of girls born using this method is (0.7547, 0.8453).