Sample Of 49 Sudden Infant Death Syndrome (SIDS) Cases ✓ Solved

A sample of 49 sudden infant death syndrome (SIDS) cases had

1. A sample of 49 sudden infant death syndrome (SIDS) cases had a mean birth weight of 2998 g. Based on other births in the county, we will assume σ = 800 g. Calculate the 95% confidence interval for the mean birth weight of SIDS cases in the county. Interpret your results.

2. True or false? Given that a confidence interval for µ is 13 + 5. a. The value of 13 in this expression is the point estimate. b. The value 5 in this expression is the estimate’s standard error. c. The value 5 in this expression is the estimate’s margin of error. d. The width of the confidence interval is 5.

3. When do we use a t-statistic instead of a z-statistic to help infer a mean?

4. Identify whether the studies described here are based on (1) single samples, (2) paired samples, or (3) independent samples. a. Cardiovascular disease risk factors are compared in husbands and wives. b. A nutritional exam is applied to a random sample of individuals. Results are compared to the results of the whole nation. c. An investigator compares vaccination histories in 30 autistic schoolchildren to a simple random sample of non-autistic children from the same school district.

5. Identify two graphical methods that can be used to compare quantitative (continuous) data between two independent groups.

6. A questionnaire measures an index of risk-taking behavior in respondents. Scores are standardized so that 100 represents the population average. The questionnaire is applied to a sample of teenage boys and girls. The data for boys is {72, 73, 86, 95, 95, 95, 96, 97, 99, 125}. The data for girls is {89, 92, 93, 98, 105, 106, 110, 126, 127, 130}. Explore the group differences with side-by-side boxplots. Create the boxplots and then describe how risk-taking behavior varies between genders.

7. Which study will require a larger sample size, one done with 80% power or 90% power when alpha (type I error) is set at 0.05 and we use the same population and expected difference and variation for both studies?

8. True or False: When using data from the same sample, the 95% confidence interval for µ will always support the results from a 2-sided, 1 sample t-test. Explain your reasoning.

Paper For Above Instructions

1. Confidence Interval for SIDS Birth Weight

To calculate the 95% confidence interval for the mean birth weight of SIDS cases in our sample, we utilize the formula for confidence intervals based on a normal distribution since we have a large sample size (n = 49) and know the population standard deviation (σ = 800 g).

The formula for a confidence interval (CI) is:

CI = μ ± Z * (σ / √n)

Where:

μ = sample mean = 2998 g

Z = Z-score corresponding to the desired confidence level (for 95%, Z ≈ 1.96)

σ = population standard deviation = 800 g

n = sample size = 49

Calculating the standard error (SE):

SE = σ / √n = 800 / √49 = 800 / 7 = 114.29 g

Now, calculating the margin of error (ME):

ME = Z SE = 1.96 114.29 ≈ 224.99 g

Now we compute the confidence interval:

CI = 2998 ± 224.99

CI = (2773.01 g, 3222.99 g)

This implies that we are 95% confident that the true average birth weight for SIDS cases in the county lies between approximately 2773 g and 3223 g.

2. True or False Analysis

a. True. The value "13" represents the point estimate or central value from which the confidence interval is calculated.

b. False. The value "5" does not represent the standard error; instead, it is the margin of error.

c. True. The value "5" is indeed the margin of error in the expression.

d. False. The width of the confidence interval is 2 times the margin of error (5), which would be 10, not 5.

3. Use of T-Statistic vs. Z-Statistic

A t-statistic is generally used instead of a z-statistic when the population standard deviation is unknown, and the sample size is small (typically n

4. Sample Types in Studies

a. The study comparing cardiovascular disease risk factors in husbands and wives uses paired samples (1) because the groups are related.

b. The nutritional exam applied to a random sample of individuals compared to the results of the whole nation uses single samples (1) as it evaluates performance based on a single cohort.

c. The comparison of vaccination histories between 30 autistic children and a random sample of non-autistic children uses independent samples (3) since the two groups are separate and unconnected.

5. Graphical Methods for Independent Groups

Two effective graphical methods that can compare quantitative data between two independent groups are:

  • Boxplots: Ideal for displaying the distribution of data across different groups allowing for easy comparison of medians and ranges.
  • Bar Charts: Useful for displaying means and standard deviations of the groups, which helps visualize the differences in central tendency.

6. Boxplots Analysis for Risk-taking Behavior

The data provided shows how teenage boys and girls score on a standardized risk-taking behavior index. The boys' data set is {72, 73, 86, 95, 95, 95, 96, 97, 99, 125} and the girls' data set is {89, 92, 93, 98, 105, 106, 110, 126, 127, 130}.

To explore differences, side-by-side boxplots can be utilized, showing the interquartile ranges and outliers of each group. The boys' scores suggest a slightly skewed distribution with an outlier at 125, while the girls' scores show a more uniform distribution with no apparent outliers.

From the boxplots, one can interpret that boys tend to have lower risk-taking behavior scores compared to girls, with a significant outlier indicating that a boy exhibited an unusually high risk-taking score.

7. Sample Size Comparison by Power

A study with 90% power will generally require a larger sample size compared to one with 80% power to detect the same effect size with a less Type II error likelihood. This is due to higher power reducing the chances of a false negative, needing more data to provide adequate evidence against the null hypothesis.

8. Confidence Interval and T-Test Relationship

True or False: When using data from the same sample, the 95% confidence interval for µ will always support the results from a 2-sided, 1 sample t-test.

This statement is true because if the confidence interval does not include the null hypothesis value during a t-test, the t-test would indicate a significant result. Likewise, if the t-test shows a non-significant result, the confidence interval will capture the null hypothesis value.

References

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