CONFIDENCE INTERVALS CALCULATION 5 Calculating Confidence Intervals ✓ Solved

The 95% and 99% confidence interval are shown in the range of a data set. This range of values shows that the chances that population means can lie in this range of interval. Calculate confidence intervals for the quantitative variables in the Heart Rate Dataset.

In the Heart Rate Dataset, we have two qualitative variables as the male and female heart rate. These two variables are categorized into two groups in which males denoted as 0, and the female is denoted as 1. The heart rate took before as resting and after exercise of the person. Here we are calculating the z-score of the data.

The z-score calculating by the formula given by, z = (x – μ) / σ. One Population Statistics Sample Statistic x-mean 15 Sample Size n 150 Sample Standard Deviation 2 Here, we have the sample of size n= 150, which is then, from the overall population size of the given data is 200, and the male population is 108, and the female population is 92. The community follows a normal distribution.

We are taking this range of delivery to calculate the one sample population. By using the data tool, we have got the value of the sample mean is 15. The estimated standard deviation is 2. Now, we have calculated the z- scores, and we have taken a hypothetical mean value where μ=15. This is exactly equal to the sample mean of the data received.

So, the calculated Z score value is 0, which shows that the benefits are exact average. By calculating we get the value of the standard error 0.1633. Here we have the table for the confidence intervals of the z-score at 95% and 99% level of confidence intervals. Confidence Intervals using z-scores.

CL 90% upper 15.27 z 1.645 lower 14.73 CL 95% upper 15.32 z 1.96 lower 14.68 CL 99% upper 15.42 z 2.576 lower 14.58 In the table, we have calculated the two-tailed test we get the lower and the upper limits of the confidence intervals. At the 95% level, the value of the Z score is 1.96 and at the 99% level, the value is 2.576.

The 95 % confidence interval for the mean is (15.32, 14.68) and 99% level of the confidence interval for the mean is (15.42, 14.58). Two Population (mean) Statistics First Sample Mean First Sample Size Second Sample Mean Second Sample Size Difference Mean Diff -1.

Now again, we are considering two samples. First sample with the size 40 and second sample with the size 50. By using the data tool, we have got the value of the sample mean of the first sample is 50 and the value of the mean of the second sample is 51. The differences between the two means are of value 1. The standard deviation of the first sample is 1.2 and the standard deviation of the second sample is 1.8.

By solving the given data, we get the z-score -163.7846. Also from the table, we get the value of the standard deviation of 0.317. The hypothetical difference between the two samples 51. So here the difference between the sample value and the hypothetical value is -52.

Here, from the table the calculated standard error. C I of Difference using z-scores.

CL 90% upper -0.478 z 1.645 lower -1.522 CL 95% upper -0.378 z 1.96 lower -1.622 CL 99% upper -0.182 z 2.576 lower -1.818 Here we have the table for the confidence intervals of the z-score at 95% and 99% level of confidence intervals. The z-score value at 95% and 99% is 1.96 and 2.576. The 95 % confidence interval for the mean is (-0.378,-1.622) and 99% level of the confidence interval for the mean is (-0.182,-1.818).

Paper For Above Instructions

Confidence intervals are statistical tools that provide a range of values that likely include a population parameter, such as the mean or proportion. In this paper, we calculate the confidence intervals for heart rates based on a dataset comprising male and female subjects. Understanding how to interpret and calculate these intervals is critical for researchers in fields such as healthcare and psychology.

To begin, we examined the heart rate dataset, which includes variables for gender and heart rate measurements before and after exercise. The analysis focused specifically on calculating confidence intervals for the means of the heart rate data segregated by gender, where males are categorized as 0 and females as 1. Using a sample size of n=150 from a larger population of 200, with 108 males and 92 females, we follow a normal distribution assumption.

Utilizing the formula for the z-score, z = (x – μ) / σ, we can process the heart rate data to determine confidence intervals effectively. In this context, our sample mean (x) is 15, with a sample standard deviation (σ) of 2. Plugging these into our z-score formula requires us to consider hypothetical values for a population mean (μ). Here, we set μ also to 15, aligning with our sample mean.

The calculated z-score is 0, indicating that our sample mean is precisely average concerning our population mean. The standard error (SE), calculated using the standard deviation and the sample size, is 0.1633. Utilizing this information, we can construct confidence intervals.

The confidence interval calculations provide a critical insight into statistical significance and reliability. Our findings reveal that at the 95% confidence level, the interval is (14.68, 15.32); at the 99% confidence level, it is (14.58, 15.42). This result suggests that we can be 95% confident that the true mean heart rate for the population lies within this interval, which is crucial for practical implications in clinical settings.

Next, we analyzed a second aspect of the heart rate dataset by comparing two independent samples. The first sample consisted of 40 subjects with a mean heart rate of 50, and the second sample comprised 50 subjects with a mean heart rate of 51. The difference between the two means was recorded as 1. The variations among the samples are essential for understanding potential differences between genders in cardiac response to exercise.

In calculating the z-score for this two-sample scenario, we determined the standard deviations of the sample means to be 1.2 and 1.8, respectively. Performing z-score calculations involves deducing standard errors from these values, leading to critical insights about the relationship between the populations. The standard error revealed from our computations is 0.317.

Confidence intervals for the difference in means can now be computed. Our results reveal a 95% confidence interval of (-1.622, -0.378), and a 99% confidence interval of (-1.818, -0.182). This indicates a significant difference between the heart rates of the two groups, suggesting that if the differences in heart rates relate directly to gender, further research and analysis would be required to explore why this variance occurs.

In conclusion, the calculations of confidence intervals for the heart rate dataset provide critical insight into the population's characteristics. By calculating z-scores and employing statistical tools to derive confidence intervals, researchers can gain a robust understanding of population traits, allowing for informed decision-making and directions for future research. Understanding these concepts is paramount in fields that rely on quantitative data to affect real-world decisions, particularly healthcare, where heart rate monitoring is integral in assessing individuals' cardiovascular health and response to exercise.

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