Question

From ring spinning section, 20 ring cops were selected randomly from two frames for testing yarn count. To know whether the frames are spinning the same count, the best statistical tools advised is

• A. Chi- square test
• B. Cluster analysis
• C. Fisherman's test
• D. Student 't' test

Hint:

• Student's t-test} is used for comparing the means of one or two groups, especially when sample sizes are small and population standard deviation is unknown.
  • One-sample t-test: Compares sample mean to a known population mean.
  • Two-sample t-test: Compares means of two independent groups (as in this problem).
  • Paired t-test: Compares means of the same group at two different times or under two different conditions.
• Chi-square test} is for categorical data (goodness-of-fit, independence) or variance tests.
• ANOVA (F-test)} is for comparing means of three or more groups. An F-test can also compare variances of two groups.
• Cluster analysis} is for grouping data, not hypothesis testing of means.

Answer 1

The Correct Option is D

The problem involves comparing the mean yarn count from two different spinning frames (samples) to determine if they are spinning the same count (i.e., if the population means of yarn count from the two frames are equal). We have:

• Two independent samples (yarn cops from two different frames).
• A continuous variable being measured (yarn count).
• The sample size for each frame is implied by "20 ring cops selected randomly from two frames". If it's 20 total (e.g., 10 from each) or 20 from each frame, it affects specifics but not the choice of test type fundamentally for small samples. Let's assume $n_1$ cops from frame 1 and $n_2$ cops from frame 2.

We want to test the hypothesis $H_0: \mu_1 = \mu_2$ (mean count from frame 1 = mean count from frame 2) against $H_1: \mu_1 \neq \mu_2$. Let's evaluate the statistical tools:

• (a) Chi-square test ($\chi^2$ test): Primarily used for:
  • Testing goodness-of-fit (e.g., if sample data fits a particular distribution).
  • Testing independence of categorical variables in a contingency table.
  • Testing hypotheses about a single population variance.
  It is not directly used for comparing means of two continuous samples.
• (b) Cluster analysis: A set of techniques used for grouping objects or cases into clusters based on their similarity. It's an exploratory data analysis tool, not a hypothesis testing tool for comparing means.
• (c) Fisherman's test (Fisher's exact test or Fisher's F-test):
  • Fisher's exact test is used for analyzing contingency tables, typically $2 \times 2$ tables with small sample sizes, to test for independence of categorical variables.
  • Fisher's F-test (ANOVA) is used for comparing means of three or more groups, or for comparing variances of two groups. If we were comparing variances of yarn count from two frames, an F-test would be appropriate. For comparing two means, a t-test is more direct.
• (d) Student 't' test: Used for hypothesis testing concerning the mean(s) of normally distributed populations when the sample size is small or population standard deviation is unknown.
  • Two-sample t-test (independent samples t-test): This is specifically designed to compare the means of two independent groups. This fits the scenario perfectly: comparing the mean yarn count from frame 1 versus frame 2.

Given the objective (to know whether the frames are spinning the same count, i.e., comparing means of two samples) and the nature of the data (yarn count, a continuous variable, likely with small to moderate sample sizes from each frame), the Student 't' test (specifically, a two-sample t-test) is the most appropriate statistical tool. \[ \boxed{\text{Student 't' test}} \]