Assignment III - PART C : Guide to ANS

PART C

1. a) Confidence Intervals (CI):

We can use the formula for a one-sample z-confidence interval:

CI = x̅ ± (z * σ / √n)

where:

• x̅ is the sample mean (145)
• z is the z-score for the desired confidence level (1.96 for 95%)
• σ is the population standard deviation (40)
• n is the sample size

Plugging in the values:

CI = 145 ± (1.96 * 40 / √60) ≈ 137.67 to 152.33

Therefore, the 95% confidence interval for the population mean is between 137.67 and 152.33.

b) Sample Size for Margin of Error (ME):

We can use the formula to find the required sample size:

n = (z * σ / ME)^2

where:

• z is the z-score for the desired confidence level (1.96 for 95%)
• σ is the population standard deviation (40)
• ME is the desired margin of error (5)

Solving for n:

n = (1.96 * 40 / 5)^2 ≈ 236.9 (check Answer with your calculator)

Since we cannot have a fractional sample size, we round up to n = 385. This means you would need a sample size of at least 385 to estimate the population mean within 5 units with 95% confidence.

2. Estimators

a) Estimator Types:

• Unbiased and Efficient: An estimator that provides the true population value on average (unbiased) and has the least variance among all unbiased estimators for a given sample size (efficient).

• Unbiased and Inefficient: An estimator that is unbiased but has a higher variance compared to an efficient estimator.

• Biased and Inefficient: An estimator that provides a skewed estimate (biased) and also has a high variance.

b) Unbiased Estimates for Sphere Diameter:

• Mean (Unbiased and Efficient):

Mean diameter = (6.33 + 6.37 + 6.36 + 6.32 + 6.37) / 5 = 6.35 cm

• Variance (Unbiased):

We can use the unbiased variance estimator:

Variance = Σ(x_i - x̅)^2 / (n-1)

where x_i is each diameter measurement and x̅ is the mean diameter.

Calculating the variance requires a bit more computation, but statistical software can help.

3. Hypothesis Testing

a) ANOVA for Two-Way Classification:

• ANOVA (Analysis of Variance) tests the null hypothesis that there are no significant differences between group means in a two-way classification design (e.g., testing the effect of fertilizer type and soil type on plant growth).

The procedure involves:

1. Defining the null and alternative hypotheses.
2. Partitioning the total variance into components explained by factors, interaction, and error.
3. Calculating the F-statistic to test the significance of each factor and interaction.
4. Interpreting the results based on p-values.

b) Sample Size using Confidence Intervals:

The sample size needed for a desired confidence interval width depends on the population standard deviation, the desired margin of error, and the confidence level. We can use the same formula from question 1b to determine the required sample size.

4. Sampling Distribution with Replacement

Population: 2, 3, 6, 8, 11

a) Mean of Population:

Mean = (2 + 3 + 6 + 8 + 11) / 5 = 6

b) Standard Deviation of Population:

You can calculate the population standard deviation using a formula or statistical software. Here's the population standard deviation formula:

σ = √(Σ(x_i - μ)^2 / N)

where:

• x_i is each value in the population
• μ is the population mean (6)
• N is the population size (5)

c) Mean of Sampling Distribution of Means:

When sampling with replacement, the mean of the sampling distribution of means is equal to the population mean itself. This is because every element in the population has an equal chance of being selected on each draw, so the sampling process doesn't introduce any bias.

d) Standard Deviation of Sampling Distribution of Means (Standard Error):

The standard deviation of the sampling distribution of means (standard error) is calculated using the formula:

σ_m = σ / √n

where:

• σ_m is the standard error
• σ is the population standard deviation (obtained from part b)
• n is the sample size

5. Sampling Distribution without Replacement

Population: 5, 10, 14, 18, 13, 24

a) Mean of Population:

Mean = (5 + 10 + 14 + 18 + 13 + 24) / 6 = 14

b) Standard Deviation of Population:

Similar to question 4b, calculate the standard deviation using the formula:

σ = √(Σ(x_i - μ)^2 / N)

where:

• x_i is each value in the population
• μ is the population mean (14)
• N is the population size (6)

c) Mean of Sampling Distribution of Means:

For samples drawn without replacement, the mean of the sampling distribution of means remains the same as the population mean (14).

d) Standard Deviation of Sampling Distribution of Means (Standard Error):

Calculating the standard error for samples without replacement requires a bit more advanced statistics. It involves a concept called the "finite population correction factor" which adjusts for the fact that elements cannot be chosen multiple times. There are specific formulas and software tools available to calculate the standard error in this case.

6. Maximum Likelihood Estimation

a) Method of Maximum Likelihood:

This is a statistical method to estimate population parameters by finding the values that maximize the likelihood function. The likelihood function expresses the probability of observing the sample data given different parameter values. The parameter values that maximize this function are considered the most likely values for the population parameters.

b) Properties of Maximum Likelihood Estimators:

• Consistency: As the sample size increases, the maximum likelihood estimator converges to the true population parameter value.
• Asymptotic Normality: Under certain conditions, the maximum likelihood estimator is approximately normally distributed for large sample sizes.
• Efficiency: In many cases, the maximum likelihood estimator is efficient, meaning it has a relatively low variance compared to other unbiased estimators.  (*add your relevant reference booke material here)

7. Sampling Theory and Hypothesis Testing

a) Two Types of Problems:

• Estimation: Aims to estimate population parameters like mean, variance, or proportion based on sample data. We discussed confidence intervals for estimation in question 1.

• Hypothesis Testing: Aims to assess the validity of a claim about a population parameter by testing a null hypothesis (no difference) against an alternative hypothesis. ANOVA is a type of hypothesis testing used to compare group means (question 3a).

b) Quality Control Example:

Here's how to analyze the production data:

1. Expected Proportion: Since the historical experience suggests 20% top quality, the expected proportion (p) in the sample is 0.2.
2. Sample Proportion: The observed proportion (q) in the sample of 400 is 50/400 = 0.125.
3. Chi-Square Test: You can perform a chi-square test to compare the observed proportion (0.125) with the expected proportion (0.2). A low p-value from the test suggests a significant difference, indicating either the sample is not representative or the 20% hypothesis might be wrong.
4. Confidence Interval for Proportion: We can also calculate a confidence interval for the true proportion of top-quality products based on the sample data using formulas for binomial proportions.

We can solve this problem using concepts of probability and confidence intervals. Here's a step-by-step breakdown:

Step 1: Analyze the Discrepancy

• Expected Top Quality: We are told that historically, 20% of the products are top quality. So, for a production of 400 articles, we would expect:

  Expected Top Quality = 20% * 400 articles = 80 articles

• Actual Top Quality: However, on the chosen day, only 50 articles were top quality. This is a significant difference compared to the expected value.

Step 2: Hypothesis Testing (Informal Approach)

While we won't perform a formal statistical test here, the large discrepancy between the expected (80) and actual (50) number of top-quality products suggests one of two possibilities:

1. Non-Representative Sample: The production on the chosen day might not be a typical representation of the overall production process. Factors like machine malfunctions or temporary changes in quality control procedures could have led to a lower percentage of top-quality products.
2. Wrong Hypothesis: The historical data might be inaccurate, or the actual percentage of top-quality products could be lower than 20%.

Step 3: Confidence Intervals for Top Quality Percentage (Informally)

Since we suspect the chosen day might not be representative, calculating a formal confidence interval wouldn't be entirely accurate. However, we can estimate a rough range for the true percentage of top-quality products based on the sample data:

• Sample Proportion: The proportion of top-quality products in the sample (one day's production) is:

  Sample Proportion = (Number of Top Quality Articles) / (Total Number of Articles) = 50 articles / 400 articles = 0.125 (12.5%)

Confidence Interval (Informal Estimation):

• We can assume a certain level of confidence (say, 95%) and estimate a range around the sample proportion (12.5%) that might hold the true population proportion with that level of confidence.

Note: This is not a statistically rigorous confidence interval due to the potential non-representativeness of the sample.

• A wider range would indicate higher uncertainty about the true proportion. Given the significant difference observed, the range might be wider than usual for a 95% confidence interval.

Example (for illustration only):

Let's hypothetically say the range could be ±5% around the sample proportion (12.5%). This would give us a very rough estimate of the confidence interval:

Lower Limit (Unreliable Estimate) ≈ 12.5% - 5% ≈ 7.5% Upper Limit (Unreliable Estimate) ≈ 12.5% + 5% ≈ 17.5%

Important Note: This is just an example to illustrate the concept. The actual confidence interval could be wider or narrower depending on the specific circumstances.

Conclusion:

The significant difference between the expected and actual number of top-quality products suggests either the chosen day was not representative or the historical data might be inaccurate. A formal statistical analysis on a larger dataset would be necessary for more precise conclusions and confidence intervals.

8. One-Way and Two-Way ANOVA

One-Way and Two-Way ANOVA (continued)

b) Two-Way ANOVA (continued):

• Analyzes the effects of two independent categorical variables (factors) and their interaction on a continuous dependent variable (e.g., testing the combined effect of fertilizer type and soil type on plant growth).
• Considers the main effects of each factor and the interaction effect between them.

Real-Life Examples:

• One-Way ANOVA:

  • Comparing the average waiting times at different bank branches.
  • Testing the effectiveness of different drugs for a particular disease.
  • Analyzing the impact of various teaching methods on student performance.

• Two-Way ANOVA:

  • Examining the combined effect of fertilizer type and soil composition on crop yield.
  • Studying the interaction between exercise intensity and duration on weight loss.
  • Investigating the influence of gender and age on voting preferences.

Procedure for One-Way and Two-Way ANOVA Tests:

Both involve similar steps:

1. Define the Hypothesis:

   • Null hypothesis (H₀): There are no significant differences between group means.
   • Alternative hypothesis (H₁): At least one group mean is different from the others.

2. Choose the ANOVA Test:

   • One-way ANOVA for comparing means of independent groups.
   • Two-way ANOVA for analyzing effects of two factors and their interaction.

3. Perform the ANOVA Test:

   • Calculate the F-statistic, a test statistic comparing the variance between groups to the variance within groups.

4. Interpret the Results:

   • Analyze the F-statistic p-value. A low p-value (typically less than 0.05) suggests rejecting the null hypothesis and concluding significant differences between group means.

5. Post-Hoc Tests (Optional):

   • If the null hypothesis is rejected, conduct post-hoc tests (e.g., Tukey's HSD) to identify which specific groups differ significantly from each other.

Additional Notes:

• Both ANOVA tests assume normality of residuals and homogeneity of variances. Tests to check these assumptions are often performed before interpreting the F-statistic.
• Data transformations might be necessary if the assumptions are violated.
• Statistical software can simplify ANOVA calculations and provide detailed output for interpretation.

I hope this comprehensive explanation clarifies the concepts behind confidence intervals, sampling distributions, hypothesis testing with ANOVA, and estimation methods. If you have any further questions or require more specific details on any of these topics, feel free to ask!