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You suspect that among fraternal twins born across the United States, the second-born twin has a higher resting heart rate from ages 0-2 years. Your instructor will provide an Excel sheet containing resting heart rate data (in beats per minute) for 129 pairs of fraternal twins, measured at age 1 year. You will be addressing the following areas, including relevant calculations and explanations; see the assignment style guide for details about what to consider in your responses:

(1)Consider how your data is obtained.

(2)Select an appropriate hypothesis test.

(3)Describe your hypothesis in mathematical and nonmathematical terms.

(4)Using your Excel sample data, complete the relevant hypothesis test calculations.

(5)Make a formal decision on your null hypothesis.

(6)Using your Excel sample data, complete the relevant confidence interval calculations.

(7)Discuss how to interpret your confidence interval.

(8)Consider the effects of increasing the sample size.

(9)Consider confounding variables

(1) Consider How Your Data Is Obtained

The dataset includes paired measurements (resting heart rates) for 129 fraternal twin pairs. Each twin pair consists of a first-born and a second-born twin, and both were measured at 1 year of age. This is observational data and not from a randomized experiment, meaning causality cannot be inferred, only association.

Because the data is collected as matched pairs, it allows control for many shared variables such as gestational environment, parenting, and socioeconomic background. This makes the paired analysis more powerful than an independent-samples approach.

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(2) Select an Appropriate Hypothesis Test

Since we are comparing two related groups (first-born and second-born twins) on a single quantitative outcome (resting heart rate), the appropriate test is a:

Paired t-test (Dependent samples t-test)

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(3) Describe Your Hypothesis in Mathematical and Nonmathematical Terms

Let

• μDmu_DμD​ be the mean difference in heart rate = (Second-born HR) – (First-born HR)

Mathematical Hypotheses:

• H0:μD=0H_0: mu_D = 0H0​:μD​=0 (No difference in mean resting heart rates)

• HA:μD>0H_A: mu_D > 0HA​:μD​>0 (Second-born twins have higher mean resting heart rates)

Nonmathematical Description:
We are testing whether second-born twins, on average, have a higher resting heart rate at age 1 than their first-born co-twin.

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(4) Perform the Hypothesis Test (Using Excel)

Steps in Excel:

1. In a new column, calculate the difference in heart rate for each pair:

   Difference = Second-born HR - First-born HR

2. Use Excel’s T.TEST function or Data Analysis ToolPak:

   • =T.TEST(array1, array2, tails, type)

   • Here, use:

     • array1 = second-born rates

     • array2 = first-born rates

     • tails = 1 (one-tailed)

     • type = 1 (paired)

OR manually compute:

• Mean difference (dˉbar{d}dˉ)

• Standard deviation of differences (s_d)

• Test statistic:

  t=dˉsd/nt = frac{bar{d}}{s_d / sqrt{n}}t=sd​/n​dˉ​

• Compare the calculated t to the critical t value from the t-distribution table with $df=n-1$ at alpha = 0.05

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(5) Make a Formal Decision on Your Null Hypothesis

Using either the p-value or the critical t-value method:

• If $p<0.05$, reject the null hypothesis: there is significant evidence that second-born twins have higher resting heart rates.

• If $p\ge 0.05$, fail to reject the null: we don’t have enough evidence of a difference.

Make sure to report:

“At the 0.05 significance level, we [reject/fail to reject] the null hypothesis…”

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(6) Confidence Interval Calculation

You can calculate a 95% confidence interval for the mean difference using:

dˉ±t∗(sdn)bar{d} pm t^* left( frac{s_d}{sqrt{n}} right)dˉ±t∗(n​sd​​)

Where:

• dˉbar{d}dˉ = mean of differences

• sds_dsd​ = standard deviation of differences

• t∗t^*t∗ = critical t-value from t-table with $df=n-1$

Use Excel:

• =CONFIDENCE.T(0.05, s_d, n) → gives margin of error

• CI = dˉ±margin of errorbar{d} pm text{margin of error}dˉ±margin of error

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(7) Interpretation of Confidence Interval

• If the entire interval is above 0, this supports your hypothesis that second-born twins tend to have higher heart rates.

• If the interval includes 0, then the mean difference might be zero, and results are not statistically significant.

“We are 95% confident that the true mean difference in resting heart rate between second-born and first-born twins lies between [lower bound] and [upper bound].”

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(8) Consider the Effects of Increasing Sample Size

Increasing the sample size (more twin pairs):

• Reduces standard error

• Increases statistical power

• Narrows the confidence interval

• Makes it more likely to detect small but real differences

This is especially useful if the effect size is small but consistent.

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(9) Consider Confounding Variables

Despite the matched-pair design, some confounding variables may still exist:

• Birth complications: Second-born twins may have had more delivery stress

• APGAR scores or early oxygen deprivation

• Order of feeding and caregiving in infancy

• Parental expectations or subtle treatment differences

• Unmeasured health issues affecting heart rate (e.g., minor infections, medications)

Control of such variables would require either a more detailed dataset or stratified analyses.

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Would you like help calculating the exact p-value or building Excel formulas for this assignment? I can also generate the confidence interval for you if you upload the dataset.

 

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