Contoh Soal Interval Keyakinan ROA: Statistika Matematika II

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Statistika Matematika II: Memahami dan Menyelesaikan Soal Interval Keyakinan ROA

Hey guys! Welcome to a deep dive into the world of statistika matematika, specifically focusing on how to tackle problems related to interval keyakinan (confidence intervals) for Return on Assets (ROA). This is a crucial concept in finance and statistics, and understanding it will definitely give you a leg up. So, let's break down a sample problem step-by-step, making sure you grasp every detail.

Contoh Soal: Menghitung Interval Keyakinan ROA

Let's get started with a real-world example. Imagine you're analyzing the banking sector in Indonesia. You know that the standard deviation of Return on Assets (ROA) for Indonesian banks is 2.35. You've taken a sample of 10 banks out of a total of 37, and this sample yields an average ROA of 0.43. Now, the big question: How can we construct a confidence interval for the population standard deviation of ROA using this information?

Here’s the problem laid out:

  • Population Standard Deviation (σ): 2.35
  • Sample Size (n): 10
  • Sample Mean (xÌ„): 0.43
  • Population Size (N): 37

Our goal is to create confidence intervals with the following confidence levels:

  • 80%
  • 90%
  • 95%

Before we dive into the calculations, it’s essential to understand why we're doing this. A confidence interval gives us a range within which we can reasonably expect the true population parameter (in this case, the standard deviation of ROA) to lie. It’s not just about getting a single point estimate; it’s about understanding the uncertainty around that estimate. This is super important in finance, where decisions are often made based on incomplete information.

Langkah-langkah Penyelesaian: A Step-by-Step Guide

Now, let’s break down the solution into manageable steps. We'll walk through each part carefully, so you can follow along and understand the logic behind it.

  1. Identify the Distribution: The first step is to recognize the appropriate statistical distribution to use. Since we are dealing with the standard deviation and a relatively small sample size, we'll use the Chi-Square (χ²) distribution. The Chi-Square distribution is commonly used for estimating confidence intervals for variances and standard deviations.

    Why Chi-Square? The Chi-Square distribution is derived from the normal distribution and is particularly useful when dealing with variances and standard deviations. It's also important to remember that the Chi-Square distribution is not symmetrical, which will affect how we calculate our confidence intervals.

  2. Determine the Degrees of Freedom: The degrees of freedom (df) are crucial for finding the correct Chi-Square values. For this problem, the degrees of freedom are calculated as the sample size minus 1: df = n - 1. In our case, df = 10 - 1 = 9.

    Understanding Degrees of Freedom: Degrees of freedom represent the number of independent pieces of information available to estimate a parameter. In simpler terms, it’s the number of values in the final calculation of a statistic that are free to vary. For a sample standard deviation, we lose one degree of freedom because we use the sample mean to calculate it.

  3. Find the Chi-Square Critical Values: Next, we need to find the Chi-Square critical values for each confidence level (80%, 90%, and 95%). These values will define the boundaries of our confidence intervals. We’ll need to look up these values in a Chi-Square distribution table or use statistical software.

    How to Use the Chi-Square Table: The Chi-Square table gives you critical values for different degrees of freedom and alpha levels (α). Alpha represents the probability of making a Type I error (rejecting the null hypothesis when it's true). For a confidence level of 95%, α = 1 - 0.95 = 0.05. Since the Chi-Square distribution is not symmetrical, we need to find two critical values: χ²_lower and χ²_upper. These correspond to α/2 and 1 - α/2, respectively.

  4. Calculate the Confidence Intervals: Finally, we'll use the Chi-Square critical values to calculate the confidence intervals for the population standard deviation. The formula for the confidence interval for the variance (σ²) is:

    ((n - 1) * s²) / χ²_upper < σ² < ((n - 1) * s²) / χ²_lower

    Where:

    • n is the sample size
    • s² is the sample variance (which is the square of the sample standard deviation)
    • χ²_upper and χ²_lower are the Chi-Square critical values

    To find the confidence interval for the standard deviation (σ), we simply take the square root of the lower and upper bounds of the variance interval.

Mencari Nilai Kritis Chi-Square (Finding Chi-Square Critical Values)

Okay, so we know we need to find these critical Chi-Square values, but how do we actually do it? Let’s walk through this process for each confidence level.

1. Confidence Level 80%

  • Alpha (α) = 1 - 0.80 = 0.20
  • α/2 = 0.10
  • 1 - α/2 = 0.90
  • Using a Chi-Square table (or software) with 9 degrees of freedom:
    • χ²(0.90, 9) ≈ 4.168
    • χ²(0.10, 9) ≈ 14.684

2. Confidence Level 90%

  • Alpha (α) = 1 - 0.90 = 0.10
  • α/2 = 0.05
  • 1 - α/2 = 0.95
  • Using a Chi-Square table (or software) with 9 degrees of freedom:
    • χ²(0.95, 9) ≈ 3.325
    • χ²(0.05, 9) ≈ 16.919

3. Confidence Level 95%

  • Alpha (α) = 1 - 0.95 = 0.05
  • α/2 = 0.025
  • 1 - α/2 = 0.975
  • Using a Chi-Square table (or software) with 9 degrees of freedom:
    • χ²(0.975, 9) ≈ 2.700
    • χ²(0.025, 9) ≈ 19.023

Perhitungan Interval Keyakinan (Calculating the Confidence Intervals)

Alright, we've got our critical values! Now comes the exciting part: plugging them into the formula and calculating our confidence intervals. Remember, we're calculating the interval for the variance first, and then we'll take the square root to get the interval for the standard deviation.

Let’s recap what we know:

  • n = 10
  • s = Sample Standard Deviation (we need to calculate this from the given Sample Mean)
  • σ = 2.35 (Population Standard Deviation)

Since we are given the population standard deviation (σ = 2.35) and the sample mean (x̄ = 0.43) but not the sample standard deviation (s), we'll proceed assuming the population standard deviation is a good estimate for our calculations. However, keep in mind that in a real-world scenario, you'd want to use the sample standard deviation if it's available.

Let's assume for the sake of demonstration that the sample standard deviation (s) is approximately equal to the population standard deviation (σ), so s ≈ 2.35. Therefore, s² ≈ 2.35² ≈ 5.5225.

Now, let’s calculate the confidence intervals for each level:

1. Confidence Level 80%

  • χ²_lower = 4.168
  • χ²_upper = 14.684
  • Variance Interval:
    • Lower Bound: ((10 - 1) * 5.5225) / 14.684 ≈ 3.388
    • Upper Bound: ((10 - 1) * 5.5225) / 4.168 ≈ 11.944
  • Standard Deviation Interval: √3.388 < σ < √11.944 ≈ 1.84 < σ < 3.46

2. Confidence Level 90%

  • χ²_lower = 3.325
  • χ²_upper = 16.919
  • Variance Interval:
    • Lower Bound: ((10 - 1) * 5.5225) / 16.919 ≈ 2.937
    • Upper Bound: ((10 - 1) * 5.5225) / 3.325 ≈ 14.957
  • Standard Deviation Interval: √2.937 < σ < √14.957 ≈ 1.71 < σ < 3.87

3. Confidence Level 95%

  • χ²_lower = 2.700
  • χ²_upper = 19.023
  • Variance Interval:
    • Lower Bound: ((10 - 1) * 5.5225) / 19.023 ≈ 2.612
    • Upper Bound: ((10 - 1) * 5.5225) / 2.700 ≈ 18.411
  • Standard Deviation Interval: √2.612 < σ < √18.411 ≈ 1.62 < σ < 4.29

Interpretasi Hasil (Interpreting the Results)

So, what do these intervals actually mean? Let's break it down:

  • 80% Confidence Interval: We are 80% confident that the true standard deviation of ROA for Indonesian banks lies between 1.84 and 3.46.
  • 90% Confidence Interval: We are 90% confident that the true standard deviation of ROA for Indonesian banks lies between 1.71 and 3.87.
  • 95% Confidence Interval: We are 95% confident that the true standard deviation of ROA for Indonesian banks lies between 1.62 and 4.29.

You'll notice that as the confidence level increases, the width of the interval also increases. This makes sense, right? The more confident we want to be, the wider the range we need to consider.

Kesimpulan (Conclusion)

Alright, guys, we've made it through a challenging problem! We've successfully calculated confidence intervals for the standard deviation of ROA using the Chi-Square distribution. Remember, the key steps are:

  1. Identify the appropriate distribution (in this case, Chi-Square).
  2. Determine the degrees of freedom.
  3. Find the Chi-Square critical values.
  4. Calculate the confidence intervals.
  5. Interpret the results.

This is a fundamental concept in statistical inference, and mastering it will be incredibly valuable in your studies and career. Keep practicing, and you'll become a pro in no time!

Stay tuned for more statistical adventures!