Consider Two Classes Of Business Statistics Suppose That

Consider Two Classes Of Business Statistics Suppose Tha

Consider two classes of business statistics. Suppose that average test score for the first class is 81 and the average test score for the second class is 78.4. What is the point estimate for the difference in the mean test scores? Answer to as many decimal places as necessary.

Suppose that average test score for the first class is 79 and the average test score for the second class is 83.7. What is the point estimate for the difference in the mean test scores? Answer to as many decimal places as necessary.

You have two samples with 50 observations each and standard deviations of 71.5 and 60.4. Find the standard error for a hypothesis test concerning the difference between two population means. Answer to two decimal places.

You have two samples with 40 observations each and standard deviations of 76.2 and 100.9. Find the standard error for testing the difference between two means. Answer to two decimal places.

Sample means are 25.4 and 31.7 with an estimated standard error of 8.1. Calculate the test statistic for testing the null hypothesis that the difference between two population means is zero. Answer to two decimal places.

Sample means are 56.5 and 45.2 with an estimated standard error of 8.8. Calculate the test statistic for testing the null hypothesis that the difference between two population means is zero. Answer to two decimal places.

In a paired sample test, the average quantities demanded in two markets are 1765 and 1688, with a standard deviation of differences of 60.5 and a sample of 17 periods. Find the standard error for the differences. Answer to two decimal places.

Using the same data as above, calculate the test statistic for the paired samples. Answer to two decimal places.

With the same data, determine the degrees of freedom for the paired samples test.

Based on the previous data, decide whether to reject the null hypothesis of equal quantities demanded at the 0.05 significance level (True or False).

Suppose the average weekly wage for women in the US is around $740 (sample size 1000, standard deviation $300), and for men it is around $900 (sample size 1000, standard deviation $450). Compute the test statistic for testing the equality of the population means. Answer to two decimal places. The number should be negative.

For a given standard error, does a larger difference between two point estimates correspond to a larger p-value when testing for the equality of means? (True or False)

The test statistic for testing the equality of two means with known standard deviations and independent samples of more than 30 observations each follows which distribution? (t, z, F, Chi square)

The test statistic for testing the equality of two means with unknown standard deviations and independent samples of more than 30 observations each follows which distribution? (t, z, F, Chi square)

The test statistic for testing the equality of two means with unknown standard deviations and matched (paired) samples follows which distribution? (t, z, F, Chi square)

The average GPA for USC sophomores is 2.8 with a standard deviation of .25, normally distributed. For a sample of 40 students, what is the probability that the sample mean exceeds 2.9?

Given the same GPA data, what is the probability a randomly selected sophomore has a GPA less than 2.9?

A lumber manufacturer’s board lengths are normally distributed with an average of 8 ft and a standard deviation of .04 ft. What are the odds that the average of a sample of 50 boards is less than 8.015 ft?

With the GPA data, for a sample size of 30, what is the value below which there is a 10% chance the sample mean will fall?

Paper For Above instruction

The collection of questions above involves fundamental concepts in business statistics, including point estimates, standard errors, hypothesis testing, t-tests, z-tests, paired sample tests, and probability calculations rooted in normal distribution theories. This comprehensive discussion aims to elucidate these concepts through detailed explanations, relevant formulas, and practical examples, thereby fostering a clearer understanding of statistical inference and its applications in business contexts.

Introduction

Statistics play a pivotal role in business decision-making, providing tools for estimating parameters, testing hypotheses, and quantifying uncertainty. The problems presented here span various aspects of inferential statistics, such as comparing means, calculating standard errors, and hypothesis testing using different distributions. A thorough grasp of these concepts enables managers and analysts to make data-driven decisions with confidence, especially in evaluating market demand, wages, or product specifications. This paper offers a comprehensive overview of these themes, incorporating detailed mathematical explanations and practical applications.

Estimating Difference Between Two Means

The initial problems examine how to find the point estimate of the difference between two population means. The concept is straightforward; it involves subtracting one sample mean from another. For example, if the average test score of the first class is 81 and that of the second class is 78.4, then the point estimate of the difference is 81 - 78.4 = 2.6. This value indicates the estimated gap in performance between the two classes.

Similarly, when the mean test scores are 79 and 83.7, the difference is 79 - 83.7 = -4.7. The negative sign indicates that the second class outperformed the first by 4.7 points. These simple calculations serve as foundational steps towards more complex inferential procedures, including confidence intervals and significance testing.

Standard Error in Comparing Two Means

Calculating the standard error (SE) of the difference between two population means is crucial for hypothesis testing. The formula for the SE when standard deviations and sample sizes are known is:

SE = sqrt[(s₁² / n₁) + (s₂² / n₂)]

where s₁ and s₂ are sample standard deviations, and n₁ and n₂ are sample sizes. For instance, with n₁ = n₂ = 50, s₁ = 71.5, and s₂ = 60.4, the SE becomes:

SE = sqrt[(71.5² / 50) + (60.4² / 50)] = sqrt[(5112.25 / 50) + (3648.16 / 50)] = sqrt[102.24 + 72.96] = sqrt[175.2] ≈ 13.24

Similarly, for n₁ = n₂ = 40, s₁ = 76.2, and s₂ = 100.9, the standard error is:

SE = sqrt[(76.2² / 40) + (100.9² / 40)] ≈ sqrt[(5802.44 / 40) + (10181.81 / 40)] ≈ sqrt[145.06 + 254.55] ≈ sqrt[399.61] ≈ 19.99

These calculations highlight how larger variability or smaller sample sizes increase the standard error, thereby affecting hypothesis testing and confidence interval widths.

Hypothesis Testing and Test Statistics

Hypothesis testing involves comparing the observed data against a null hypothesis, typically stating no difference (null hypothesis: μ₁ - μ₂ = 0). The test statistic quantifies the deviation of the observed difference from zero, scaled by the standard error:

z = (x̄₁ - x̄₂ - Δ₀) / SE

where Δ₀ is the hypothesized difference (often zero). For example, if the sample means are 25.4 and 31.7, with SE = 8.1, the test statistic is:

z = (25.4 - 31.7 - 0) / 8.1 ≈ -6.3 / 8.1 ≈ -0.78

These statistics are then compared to critical values, depending on the significance level, to determine whether to reject the null hypothesis.

Paired Samples and Differences

In paired or matched samples, the focus is on the differences within each pair. The mean difference, along with its standard deviation, allows for inference about the population mean difference. For example, with 17 pairs, a mean difference of 1765 - 1688 = 77, and a standard deviation of 60.5, the standard error of the mean difference is:

SE = 60.5 / sqrt(17) ≈ 60.5 / 4.123 ≈ 14.70

The t-statistic is calculated as:

t = (mean difference - hypothesized difference) / SE

Assuming null hypothesis of zero difference, the t-value would be:

t = 77 / 14.70 ≈ 5.24

Degrees of freedom for this test are typically n - 1, so 16 in this case.

Probability and Normal Distribution

The problems involving probabilities and GPA calculations utilize properties of the normal distribution. For the GPA example, knowing the population mean (μ = 2.8), standard deviation (σ = 0.25), and sample size (n), the sampling distribution of the mean is normal with standard error:

SE = σ / sqrt(n)

For a sample size of 40, the standard error is 0.25 / sqrt(40) ≈ 0.0395. To find the probability that the sample mean exceeds 2.9, we calculate the z-score:

z = (2.9 - 2.8) / 0.0395 ≈ 2.53

The probability that z exceeds 2.53 corresponds to approximately 0.0057, indicating a very low likelihood.

Conclusion

The concepts explored—including difference estimation, standard errors, hypothesis testing, paired samples, and probabilities—are fundamental to business statistics. Mastery of these methods allows analysts to infer population parameters, assess hypotheses, and make sound business decisions based on data. Understanding the formulas, assumptions, and interpretation of results is essential for accurate statistical analysis and effective communication of findings in a business environment.

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