Question

The viscosity of a liquid detregent is supposed to average 800 centistokes at 250C. A random sample of 10 bathces of detergent is collected and the average viscosity is 812. Suppose we know that the standard deviation of viscosity is 20 centistokes.

 Test a hypothesis with a fixed significance level of 0.05. (7 step procedure outlined in class).

 What is the smallest level of significance at which the null hypothesis will be rejected? Based upon this value, state your conclusions on the hypothesis test.

 Construct a 95% two sided CI on mean viscosity and state your conclusions on the hypothesis test.

Answer 1

1. Step 1:The null hypothesis

800

HO :μ =

Step 2: The alternative hypothesis

Ha:μ
800

Step 3: Test : t test (two tailed) for mean

Test statistic

X - u t=sn

df = n- 1= 10-1 =9

$\dpi{100} \alpha =0.05$

Critical value of t , tc = 2.262 ( from critical value table for $\dpi{100} \alpha =0.05$ , df =9)

Reject H0: if value of test statistic > 2.262

Step 4: Calculating sample statistic

812

$\dpi{100} \sigma =20$ (population standard deviation)

We shall put , s = 20

Step 5: Calculation of Test statistic

X - u t=sn

  = $\dpi{100} \frac{812-800}{20 /\sqrt{10}}$

= 1.90

Step 6: Comparison

Calculated value of t < 2.262

Step 7 : Conclusion

Fail to Reject H0

There is not sufficient evidence to conclude that average viscosity of liquid detergent is not 800.

2.t =1.90

df = n-1 =10-1 =9

P value = 0.0899

Note : excel formula for P value "=T.DIST.RT(1.90,9)"

As P value = 0.0899

We know that P value is the smallest significance level at which null hypothesis is rejected

Thus the null hypothesis will be rejected at significance level of 0.0899

Conclusion:

At 8.99% significance level , there is not sufficient evidence to conclude that average viscosity of liquid detergent is not 800.

3. The 95% confidence interval for mean is

X te *s/vn

For 95% confidence with df =9 , tc = 2.262

95% confidence interval for population mean

$\dpi{100} 812\pm 2.262*20 /\sqrt{10}$

$\dpi{100} = 812\pm14.3061$

= (797.69 , 826.31)

We can see that the 95% confidence interval include 800

Note : 95% confidence = (1-0.05)% confidence

Conclusion :

At 0.05 significance level , there is not sufficient evidence to conclude that average viscosity of liquid detergent is not 800.