An obstetrician maintains that preterm babies (gestation less than 37 weeks) have a higher variability of birth weight than do full-term babies (gestation 37-41 weeks). According to the National Vital Statistics Report, Vol. 48, No. 3, the birth weights of full time babies are normally distributed with a standard deviation of 505.6 grams. A random sample of 41 preterm babies results in a standard deviation of 840 grams. Test the obstetrician's claim that the variability in birth weight for preterm babies is more than that of full-term babies, at the level of significance 0.01.
Any help on figuring this out would be greatly appreciated. Thanks for your time.
Use my information from your previous posts.
To test the obstetrician's claim, we need to perform a hypothesis test comparing the variability in birth weight for preterm babies versus full-term babies.
Step 1: State the Hypotheses
- The null hypothesis (H0): The variability in birth weight for preterm babies is the same as that of full-term babies.
- The alternative hypothesis (Ha): The variability in birth weight for preterm babies is more than that of full-term babies.
Step 2: Set the Significance Level
The significance level, denoted as α (alpha), is given as 0.01 in this case. It represents the probability of rejecting the null hypothesis when it is actually true.
Step 3: Calculate the Test Statistic
In this case, since we are comparing the variability (standard deviations) of two independent samples, we can use the F-test statistic. The formula for this test statistic is:
F = (s1^2) / (s2^2)
where s1^2 is the variance of the first sample (preterm babies) and s2^2 is the variance of the second sample (full-term babies).
Step 4: Determine the Critical Value
To determine the critical value for the test, we need to use the F-distribution table (or statistical software). The critical value is the value that separates the rejection region (where we reject the null hypothesis) from the non-rejection region (where we fail to reject the null hypothesis).
The critical value depends on the degrees of freedom of the two samples. In this case, since the numerator degrees of freedom (df1) is the number of preterm babies minus 1 (n1 - 1) and the denominator degrees of freedom (df2) is the number of full-term babies minus 1 (n2 - 1). Substituting the values, df1 = 41 - 1 = 40 and df2 = 0 (since n2 is not provided), we can find the corresponding critical value of F.
Step 5: Calculate the P-value
The P-value is the probability of obtaining a test statistic more extreme than the observed value, assuming the null hypothesis is true. Since our alternative hypothesis is one-tailed (claiming more variability), we need to find the probability in the upper tail of the F-distribution, given our test statistic and degrees of freedom.
Step 6: Make the Decision
Compare the calculated P-value with the significance level (α) to make a decision. If the P-value is less than α, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
Let's perform the calculations:
Given:
Standard deviation of full-term babies (s2) = 505.6 grams
Standard deviation of preterm babies (s1) = 840 grams
Sample size of preterm babies (n1) = 41
Calculations:
Test statistic:
F = (s1^2) / (s2^2) = (840^2) / (505.6^2)
Degrees of freedom:
df1 = n1 - 1 = 41 - 1 = 40
df2 = n2 - 1
Critical value:
Using statistical software or an F-distribution table (with df1 = 40 and df2 = 0), find the critical value for a significance level of 0.01.
P-value:
Calculate the upper tail probability using the F-distribution table or statistical software with the test statistic and degrees of freedom.
Make the Decision:
Compare the calculated P-value with the significance level (0.01). If the P-value is less than 0.01, reject the null hypothesis. Otherwise, fail to reject the null hypothesis.
Note: It is mentioned in the given information that the sample size of full-term babies (n2) is not provided. This information is necessary to complete the hypothesis test accurately.