Elementary Statistics and Computer Application

Test for single Mean

1.Form the null hypothesis

• Ho: µ=µ_o
• (i.e) There is no significance difference between the sample mean and the population mean
2.Form the Alternate hypothesis

• H1 : µ≠µ_o (or µ>µ_o or µ<µ_o)
• ie., There is significance difference between the sample mean and the population mean
3. Level of Significance

• The level may be fixed at either 5% or 1%
4. Test statistic
which follows t distribution with (n-1) degrees of freedom

6.Find the table value of t corresponding to (n-1) d.f. and the specified level of significance.

7.Inference

• If t < ttab we accept the null hypothesis H0. We conclude that there is no significant difference sample mean and population mean (or) if t > ttab we reject the null hypothesis H0. (ie) we accept the alternative hypothesis and conclude that there is significant difference between the sample mean and the population mean
Example 1

• Based on field experiments, a new variety of green gram is expected to given a yield of 12.0 quintals per hectare. The variety was tested on 10 randomly selected farmer’s fields. The yield (quintals/hectare) were recorded as 14.3,12.6,13.7,10.9,13.7,12.0,11.4,12.0,12.6,13.1. Do the results conform to the expectation?
Solution

• Null hypothesis H0: µ=12.0
• (i.e) the average yield of the new variety of green gram is 12.0 quintals/hectare.
• Alternative Hypothesis: H1:µ≠ 12.0
• (i.e) the average yield is not 12.0 quintals/hectare, it may be less or more than 12 quintals / hectare
• Level of significance : 5 %
Test statistic



From the given data







= 1.0853



Now



• Table value for t corresponding to 5% level of significance and 9 d.f. is 2.262 (two tailed test)

Inference:

• t < ttab
• We accept the null hypothesis H0
• We conclude that the new variety of green gram will give an average yield of 12 quintals/hectare.
• Note
• Before applying t test in case of two samples the equality of their variances has to be tested by using F-test


or



• where is the variance of the first sample whose size is n1.
• is the variance of the second sample whose size is n2.
• It may be noted that the numerator is always the greater variance. The critical value for F is read from the F table corresponding to a specified d.f. and level of significance

Inference

• F <Ftab
• We accept the null hypothesis H₀.(i.e) the variances are equal otherwise the variances are unequal.