Daily Archives: May 27, 2015


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If  is true then:



This is our test statistic.

We reject H0 if the calculated value of our test statistic is less than -zα/2 or greater than +zα/2 (i.e., if it takes a value sufficiently far out in the tails of the standard normal distribution for us to think  is unlikely to be true).



The weights of fish in an aquaculture pond are considered to be normally distributed with a mean of 3.1Kg and a standard deviation of 1.1Kg. A random sample of size 30 is selected from the pond and the sample mean is found to be 2.37Kg. Is there sufficient evidence to indicate that the mean weight of the fish differs from 3.1Kg? Use a 10 level of significance.

hypoteses z test example





 Conclusion: The mean weight of the fish differs from 3.1Kg (at the 10% level of significance).






Hypothesis Testing

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  • A hypothesis is a statement (assumption) about a population parameter
    • population mean (Example: The mean monthly cell phone bill of this city is  μ = $42)
    • population proportion (Example: The proportion of adults in this city with cell phones is  π = 0.68)
  • Null Hypothesis
    • The hypothesis that assumes the status quo – that the old theory, method or standard is still true; the complement of the alternative hypothesis
    • Always contains ‘=‘ , ‘≤’ or ‘³’ sign
    • May or may not be rejected
    • Is always about a population parameter, ,not about a sample statistic
  • Alternative Hypothesis
    • The hypothesis that complements the null hypothesis.
    • Usually it is the hypothesis that the researcher is interested in proving
  • The Null and Alternative Hypotheses are mutually exclusive
    • e. only one of them can be true
  • The Null Hypothesis is assumed to be true
  • The burden of proof falls on the Alternative Hypothesis
  • Example: investigate if the mean monthly cell phone bill is $42
    • H0: μ = 42
    • H1: μ ≠ 42

Level of Significance and rejection region












Steps for the hypothesis test…

  1. State the null hypothesis, H0 and the alternative hypothesis, H1
  2. Choose the level of significance, a, and the sample size, n
  3. Determine the appropriate test statistic and sampling distribution
  4. Determine the critical values that divide the rejection and non-rejection regions
  1. Collect data and compute the value of the test statistic
  2. Make the statistical decision and state the managerial conclusion
  • If the test statistic falls into the non-rejection region, do not reject the null hypothesis H0.
  • If the test statistic falls into the rejection region, reject the null hypothesis
  • Express the managerial conclusion in the context of the real-world problem


  • p-value: Probability of obtaining a test statistic more extreme ( ≤ or ³ ) than the observed sample value given H0 is true
    • Also called observed level of significance
    • Smallest value of a  for which H0 can be rejected
    • Obtain the p-value from a table or computer
  • If p-value  <  a ,  reject H0
  • If p-value  ³  a ,  do not reject H0

Two populatio means tail test












Rules to follow:


Decision Rule:

Test Statistic:



Normal Distributions

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Normal Distributions

  • Also known as the Z distribution
    • Mean is 0
    • Standard deviation is 1
  • Characteristics of a Normal Distribution
    • Continuous Random Variable
    • - ∞ < x < + ∞
    • Curve is symmetrical around the mean (m).
    • Area under curve = 1
    • Mean & standard deviation uniquely determine a normal distribution.

Normal Distribution




value of Z




To find  P(a < X < b)  when  X  is distributed normally:

  1. Draw the normal curve for the problem in terms of X
  2. Translate X-values to Z-values and put Z values on your diagram
  3. Use the Standardised Normal Table

Example: Suppose X is normally distributed with mean 8 and std dev 5. Find P(X < 8.6)

Z value example






Z value table









Finding the X for a Known Probability:


  1. Draw a normal curve placing all known values on it such as mean of X and Z
  2. Shade in area of interest and find cumulative probability
  3. Find the Z value for the known probability
  4. Convert to X units using the formula:


How Large is Large Enough?

  • For most population distributions, n ≥ 30 will give a sampling distribution that is nearly normal
  • For fairly symmetric population distributions, n ≥ 5 is sufficient
  • For normal population distributions, the sampling distribution of the mean is always normally distributed



  • A point estimate is the value of a single sample statistic
  • A confidence interval provides a range of values constructed around the point estimate

Point Estimates







Confidence Level  (1-a)

  • Common confidence levels = 90%, 95% or 99%
    • Also written (1 – a) = 0.90, 0.95 or 0.99
  • A relative frequency interpretation
    • In the long run, 90%, 95% or 99% of all the confidence intervals that can be constructed (in repeated samples) will contain the unknown true parameter
  • For example, if we were to randomly select 100 samples and use the results of each sample to construct 95% confidence intervals, approximately 95 out of 100 would contain the population mean

 Confident Interval for Population when standard deviation is known












Finding the critical value Z











So, what happens if we don’t know the standard deviation of the population????

  • If the population standard deviation σ  is unknown, we can substitute the sample standard deviation, S
  • This introduces extra uncertainty, since S is variable from sample to sample
  • So we use the t distribution instead of the normal distribution

Confidence Interval Estimate:

Interval estimate



where t is the critical value of the t distribution with n -1 degrees of freedom and an area of α/2 in each tail

t distribution increasing










A random sample of n = 25 has X = 50 & S = 8.

Form a 95% confidence interval for μ:

d.f. = n – 1 = 24,  so

example of distribution





The confidence interval is  46.698 ≤ μ ≤ 53.302


Required Sample Size Example

If s = 45, what sample size is needed to estimate the mean within ± 5 with 90% confidence?

example 2



So the required sample size is n = 220