c) In order to investigate the effect of gender on the birth weight of the baby, birth weight information of 56 babies was obtained by sampling according to the classification among the birth information cards of babies born in 2016.
baby boys baby girls
Sample mean (Ẋ) 3.250 kg 3.400 kg
Sample standard deviation (S) 0.400 0.500
Number of observations (n) 28 28
Since it is known that it fits the normal distribution in both distributions, is there any difference between these two groups in terms of gender? First write the hypothesis and test it at the error level of α= 0.05 and interpret the results.
Hypothesis:
H0: There is no difference between the birth weights of baby boys and baby girls.
Ha: There is a difference between the birth weights of baby boys and baby girls.
Test statistic:
t = (Ẋ1 - Ẋ2) / (S1^2/n1 + S2^2/n2)^0.5
t = (3.250 - 3.400) / (0.400^2/28 + 0.500^2/28)^0.5
t = -0.735
At the error level of α= 0.05, the critical value of t is 1.671. Since the calculated t value is lower than the critical value, we fail to reject the null hypothesis. This means that there is no statistically significant difference between the birth weights of baby boys and baby girls.
smort
To test whether there is a difference in birth weight between baby boys and baby girls, we can conduct a two-sample t-test. The null hypothesis (H0) is that there is no difference in the mean birth weights between the two groups, while the alternative hypothesis (Ha) is that there is a difference.
H0: μboys = μgirls (where μ represents the population mean birth weight of baby boys and baby girls, respectively)
Ha: μboys ≠ μgirls
Since we have the sample means, standard deviations, and number of observations for both groups, we can calculate the test statistic and compare it to the critical value.
The test statistic t is given by:
t = (Ẋboys - Ẋgirls) / √[(Sboys^2 / nboys) + (Sgirls^2 / ngirls)]
Where:
Ẋboys = sample mean birth weight of baby boys
Ẋgirls = sample mean birth weight of baby girls
Sboys = sample standard deviation birth weight of baby boys
Sgirls = sample standard deviation birth weight of baby girls
nboys = number of observations for baby boys
ngirls = number of observations for baby girls
Plugging in the values:
t = (3.250 - 3.400) / √[(0.400^2 / 28) + (0.500^2 / 28)]
Calculating t:
t = -0.150 / √[(0.016 + 0.020)]
Calculating the pooled standard deviation:
Sp = √[((nboys - 1) * Sboys^2 + (ngirls - 1) * Sgirls^2) / (nboys + ngirls - 2)]
Sp = √[((28 - 1) * 0.400^2 + (28 - 1) * 0.500^2) / (28 + 28 - 2)]
Calculating the degrees of freedom:
df = nboys + ngirls - 2
df = 28 + 28 - 2
Using the calculated t and degrees of freedom, we can now compare the test statistic to the critical value.
Next, we would consult a t-distribution table or use statistical software to find the critical value for a t-test with the given degrees of freedom and significance level (α = 0.05). If the absolute value of the calculated t is greater than the critical value, we reject the null hypothesis and conclude that there is a statistically significant difference in birth weight between baby boys and baby girls. If the absolute value of the calculated t is less than or equal to the critical value, we fail to reject the null hypothesis, indicating that there is not enough evidence to conclude a difference in birth weight between the two groups.
Finally, we would interpret the results in terms of the hypothesis test and discuss the implications of the findings for the effect of gender on birth weight.
To investigate the effect of gender on the birth weight of the baby, we can conduct a hypothesis test.
The hypotheses for this test would be as follows:
Null Hypothesis (H0): There is no difference between the mean birth weight of baby boys and baby girls.
Alternative Hypothesis (Ha): There is a difference between the mean birth weight of baby boys and baby girls.
To test these hypotheses, we can use a two-sample t-test since we have two independent samples and we want to compare the means of two populations.
The test statistic for the two-sample t-test is given by:
t = (Ẋ1 - Ẋ2) / sqrt[(S1^2/n1) + (S2^2/n2)]
Where:
Ẋ1 and Ẋ2 are the sample means for baby boys and baby girls respectively,
S1 and S2 are the sample standard deviations for baby boys and baby girls respectively,
n1 and n2 are the number of observations for baby boys and baby girls respectively.
Let's calculate the test statistic:
For baby boys:
Ẋ1 = 3.250 kg, S1 = 0.400 kg, n1 = 28
For baby girls:
Ẋ2 = 3.400 kg, S2 = 0.500 kg, n2 = 28
Now we can calculate the test statistic using the formula mentioned above.
t = (Ẋ1 - Ẋ2) / sqrt[(S1^2/n1) + (S2^2/n2)]
t = (3.250 - 3.400) / sqrt[(0.400^2/28) + (0.500^2/28)]
After calculating the value of t, we can compare it with the critical value from the t-distribution table at the chosen significance level (α = 0.05) and degrees of freedom (df = n1 + n2 - 2).
If the calculated t value is greater than the critical value, we reject the null hypothesis. If not, we fail to reject the null hypothesis.
Interpreting the results would depend on the outcome of the hypothesis test. If we reject the null hypothesis, it would indicate that there is a statistically significant difference between the mean birth weights of baby boys and baby girls. On the other hand, if we fail to reject the null hypothesis, it would suggest that there is no significant difference between the mean birth weights of baby boys and baby girls.