Question

Question 3. (12 marks) It is posited that babies born at different times of the year may develop the ability to crawl at different ages. Thirty two babies born in January crawled at an average age of 29.75 weeks, with a standard deviation of 7 weeks. Another 32 babies born in August crawled at an average of 33.5 weeks with a standard deviation of 7 weeks. Test whether the posited claim is true. Use α=0.05. Use RStudio or R on Jupyter in order to solve this problem. Provide all your work with your solutions. How valid is the test you conducted? In order to answer this question List all assumptions required. For each assumption, indicate whether it holds and why it does or does not. Can you provide graphical displays in order to assess a particular assumption? What would you need in order to produce such graphs?

Answer 1

Solution:-

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis: u₁ = u ₂
Alternative hypothesis: u₁$\neq$ u ₂

Note that these hypotheses constitute a two-tailed test.

Formulate an analysis plan. For this analysis, the significance level is 0.05. Using sample data, we will conduct a two-sample t-test of the null hypothesis.

Analyze sample data. Using sample data, we compute the standard error (SE), degrees of freedom (DF), and the t statistic test statistic (t).

SE = sqrt[(s₁²/n₁) + (s₂²/n₂)]
SE = 1.75
DF = 60
t = [ (x₁ - x₂) - d ] / SE

t = - 2.143

where s₁ is the standard deviation of sample 1, s₂ is the standard deviation of sample 2, n₁ is the size of sample 1, n₂ is the size of sample 2, x₁ is the mean of sample 1, x₂ is the mean of sample 2, d is the hypothesised difference between the population means, and SE is the standard error.

Since we have a two-tailed test, the P-value is the probability that a t statistic having 60 degrees of freedom is more extreme than -2.143; that is, less than -2.143 or greater than 2.143.

Thus, the P-value = 0.036.

Interpret results. Since the P-value (0.036) is less than the significance level (0.05), we cannot accept the null hypothesis.

From the above test we have sufficient evidence in the favor of the claim that babies born at different times of the year may develop the ability to crawl at different ages.