Study Notes on Single Sample Z-Test and Confidence Intervals

Introduction to Inferential Statistics

• In this unit, we focus on statistical tests for examining research predictions through sample data analysis.

• The primary focus is on inferential statistics, specifically single sample hypothesis tests.

• Emphasis will be placed on the single sample z-test about the mean.

Single Sample Z-Test

Definition

• The single sample z-test is used to determine if the mean value of a variable measured in one sample significantly differs from an established mean value from a population or theory.

Requirements for Use

• Knowledge of the population standard deviation (C3) of the variable being measured.

• Alternatively, the sample size must be 30 or greater.

• These criteria can be satisfied either separately or together.

Research Design

• Researchers obtain a random sample and measure a single variable of interest.

• The mean of the sample is calculated and compared against a predetermined typical value.

Example Problem 1

Scenario

• Population mean engagement score = 100 (normally distributed).

• A training program claims to increase engagement scores; thus, we conduct a hypothesis test.

• 40 randomly selected employees participate; their mean engagement score = 104, standard deviation = 12.

Hypotheses

1. Null Hypothesis (H₀): BC (population mean) = 100

2. Alternative Hypothesis (H₁): BC > 100 (one-tailed test, greater than).

Rejection Region and Critical Value

• Decision Rule: Reject H₀ if z obtained > critical z value.

• Alpha level: 0.05 (5% significance level).

• Rejection region is in the upper tail of the normal distribution.

• Critical z value calculation:

  • Area in rejection region = 0.05, resulting in a proportion of 0.45 in the non-rejection region.

  • Critical value corresponding to 0.45 is z = 1.645.

Z Obtained Value Calculation

• Formula: $z<em>{obtained} = \frac{\bar{x} - \mu</em>0}{\sigma_{x}/\sqrt{n}}$

  • Sample mean ($\bar{x}$) = 104

  • Null mean ($\mu_0$) = 100

  • Sample standard deviation ($\sigma_{x}$) = 12;

  • Sample size ($n$) = 40.

• Calculation:

  • $z_{obtained} = \frac{104 - 100}{12/\sqrt{40}} = \frac{4}{1.897} \approx 2.11$

Conclusion

• Since 2.11 > 1.645, we reject the null hypothesis.

• There is evidence supporting that employees score significantly higher engagement after the training program.

Example Problem 2

Scenario

• Repeating the engagement study but this time the claim is that the new program may decrease engagement.

• 60 employees participated; their mean score = 96.5, standard deviation = 10.25.

Hypotheses

1. Null Hypothesis (H₀): BC = 100

2. Alternative Hypothesis (H₁): BC < 100 (one-tailed test, less than).

Rejection Region and Critical Value

• Alpha = 0.01; critical value corresponds to negative z.

  • Finding z for rejection region (0.01 alpha): critical value = -2.33.

Z Obtained Value Calculation

• $z_{obtained} = \frac{96.5 - 100}{10.25/\sqrt{60}} = \frac{-3.5}{1.32} \approx -2.64$

Conclusion

• z = -2.64 < -2.33, reject null hypothesis. There is evidence that engagement scores are significantly lower.

Example Problem 3

Scenario

• 35 employees underwent a new program with an average engagement score of 98.

Hypotheses

1. Null Hypothesis (H₀): BC = 100

2. Alternative Hypothesis (H₁): BC differs from 100 (two-tailed).

Rejection Regions and Critical Values

• Total alpha = 0.05; each tail has 0.025.

  • Critical values = -1.96 and 1.96.

Z Obtained Value Calculation

• $z_{obtained} = \frac{98 - 100}{11.2/\sqrt{35}} = \frac{-2}{1.89} \approx -1.06$

Conclusion

• z = -1.06 is within the non-rejection region. Do not reject null hypothesis. No support for the training program's effect on engagement scores.

Confidence Intervals

Definition

• Confidence intervals estimate a true population value within a specific range and degree of probability.

Importance

• Provides researchers with a better sense of where the true population mean may fall based on sample data.

Calculation of Confidence Interval

Formula

$CI = \bar{x} \pm z \times \left(\frac{\sigma}{\sqrt{n}}\right)$

• Where:

  • $\bar{x}$ = sample mean

  • $\sigma$ = population standard deviation

  • $n$ = sample size

  • z = critical z value (two-tailed).

Example Problem 4

Scenario

• A study reports the average commute time for 100 employees = 28 minutes; population standard deviation = 4 minutes.

Calculation

1. Sample mean, $\bar{x} = 28$; $\sigma = 4$; $n = 100$.

2. Critical z value at alpha = 0.05 = ±1.96.

3. Margin of error calculation:

   • $ME = z \times \left(\frac{\sigma}{\sqrt{n}}\right) = 1.96 \times \left(\frac{4}{\sqrt{100}}\right) = 1.96 \times 0.4 = 0.784$

4. Confidence Interval:

   • Lower bound = 28 - 0.784 = 27.216

   • Upper bound = 28 + 0.784 = 28.784

   • Thus, $CI = [27.216, 28.784]$.

Conclusion

• We estimate that the true mean time for employees to return home from work is between 27.216 and 28.784 minutes, assuming an alpha of 0.05.