Can someone help me start this please:
There is some indication in medical literature that doctors may have become more aggressive in
inducing labor or doing preterm cesarean sections when a woman is carrying twins. Records at a large hospital show that of the 43 sets of twins born in 1990, 20 were delivered before the 37th week of pregnancy. In 2000, 26 of 48 sets of twins were born preterm. Does this indicate an increase in the incidence of early births of twins? Test an appropriate hypothesis and state your conclusion. Calculate and interpret a 90% confidence interval.
To test whether there is an increase in the incidence of early births of twins, we can set up a hypothesis test. Let's define the following:
p₁ = proportion of twins born preterm in 1990
p₂ = proportion of twins born preterm in 2000
The null hypothesis (H₀) states that there is no change in the incidence of early births of twins, i.e., p₁ - p₂ = 0.
The alternative hypothesis (H₁) states that there is an increase in the incidence of early births of twins, i.e., p₁ - p₂ > 0.
To test these hypotheses, we can perform a hypothesis test for the difference in proportions.
Step 1: Calculate the sample proportions
In 1990, out of 43 sets of twins, 20 were born preterm: p₁ = 20/43 ≈ 0.465.
In 2000, out of 48 sets of twins, 26 were born preterm: p₂ = 26/48 ≈ 0.542.
Step 2: Calculate the standard error
The standard error (SE) for the difference in proportions can be calculated using the following formula:
SE = sqrt[(p₁ * (1 - p₁))/n₁ + (p₂ * (1 - p₂))/n₂],
where n₁ and n₂ are the sample sizes.
SE = sqrt[(0.465 * (1 - 0.465))/43 + (0.542 * (1 - 0.542))/48]
≈ 0.102
Step 3: Calculate the test statistic
The test statistic can be calculated using the formula:
z = (p₁ - p₂) / SE.
z = (0.465 - 0.542) / 0.102
≈ -0.755
Step 4: Calculate the p-value
The p-value can be determined by finding the area under the standard normal distribution curve to the right of the test statistic (-0.755). Using a standard normal distribution table or a statistical software, we find that the p-value is approximately 0.775.
Step 5: Make a decision
Since the p-value (0.775) is greater than the significance level (α = 1 - confidence level), we fail to reject the null hypothesis. There is not enough evidence to conclude that there is an increase in the incidence of early births of twins.
Step 6: Calculate the confidence interval
To calculate the confidence interval, we can use the formula:
CI = (p₁ - p₂) ± z * SE,
where z is the critical value corresponding to the desired confidence level.
For a 90% confidence level, the critical value is approximately 1.645.
CI = (0.465 - 0.542) ± 1.645 * 0.102
≈ (-0.153, 0.080)
Interpretation: We are 90% confident that the true difference in proportions of twins born preterm between 1990 and 2000 lies between -0.153 and 0.080. Since the confidence interval contains zero, it suggests that there is no statistically significant difference in the incidence of early births of twins between these two years.
To determine if there is an increase in the incidence of early births of twins, we can test the following hypothesis:
Null hypothesis (H0): The proportion of preterm births of twins has remained the same.
Alternative hypothesis (Ha): The proportion of preterm births of twins has increased.
To test this, we can use a hypothesis test for the difference in proportions. Here are the steps to conduct the hypothesis test:
1. Calculate the proportions of preterm births for each year:
- Proportion of preterm births in 1990: 20 / 43 = 0.465
- Proportion of preterm births in 2000: 26 / 48 = 0.542
2. Calculate the pooled proportion:
- Pooled proportion = (number of preterm births in 1990 + number of preterm births in 2000) / (total number of twins in 1990 + total number of twins in 2000)
- Pooled proportion = (20 + 26) / (43 + 48)
3. Calculate the standard error:
- Standard error = sqrt((pooled proportion * (1 - pooled proportion) / (n1 + n2))
- n1 = total number of twins in 1990 = 43
- n2 = total number of twins in 2000 = 48
4. Calculate the test statistic:
- Test statistic = (proportion of preterm births in 2000 - proportion of preterm births in 1990) / standard error
5. Determine the critical value for a 90% confidence interval based on the test statistic and the degrees of freedom. The degrees of freedom can be calculated using the formula: df = n1 + n2 - 2.
6. Compare the test statistic to the critical value. If the test statistic is greater than the critical value, reject the null hypothesis. Otherwise, fail to reject the null hypothesis.
7. State the conclusion based on the hypothesis test.
To calculate and interpret a 90% confidence interval, use the following steps:
1. Calculate the margin of error:
- Margin of error = critical value * standard error
2. Calculate the lower and upper limits of the confidence interval:
- Lower limit = (proportion of preterm births in 2000 - proportion of preterm births in 1990) - margin of error
- Upper limit = (proportion of preterm births in 2000 - proportion of preterm births in 1990) + margin of error
3. Interpret the confidence interval by stating that you are 90% confident that the true difference in proportions lies within the calculated interval.
Performing these steps will allow you to test the hypothesis and analyze the data to draw valid conclusions about the incidence of early births of twins.