Question

The concentration of active ingredient in a liquid laundry detergent is thought to be affected by the type of catalyst used in the process. The standard deviation of active concentration is known to be 3 grams per liter, regardless of the catalyst type. Ten observations on concentration are taken with each catalyst, and the data follow:

Catalyst 1: 57.5, 66.4, 65.4, 65.4, 65.2, 62.6, 67.6, 63.7, 67.2, 71.0

Catalyst 2: 67.0, 72.4, 70.3, 69.3, 64.8, 69.6, 68.6, 69.4, 65.3, 68.8

(a) Find a 95% two-sided confidence interval on the difference in mean active concentrations for the two catalysts. Find the P-value.

Answer 1

The 95% confidence interval would be given by
`-5.980 \leq \mu_1 -\mu_2 \leq -0.720`

`p_v =2*P(Z<-2.497)=0.012`

Explanation:

Previous concepts

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".

The margin of error is the range of values below and above the sample statistic in a confidence interval.

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

`\bar X_1 =65.2` represent the sample mean 1

`\bar X_2 =68.55` represent the sample mean 2

n1=10 represent the sample 1 size

n2=10 represent the sample 2 size

`s_1 =3.55` sample standard deviation for sample 1

`s_2 =2.29` sample standard deviation for sample 2

`\sigma =3` represent the population standard deviation

`\mu_1 -\mu_2` parameter of interest.

Part a

The confidence interval for the difference of means is given by the following formula:

`(\bar X_1 -\bar X_2) \pm z_(\alpha/2)\sqrt{\sigma^2((1)/(n_1)+(1)/(n_2))}` (1)

The point of estimate for
`\mu_1 -\mu_2` is just given by:

`\bar X_1 -\bar X_2 =65.2-68.55=-3.35`

Since the Confidence is 0.95 or 95%, the value of
`\alpha=0.05` and
`\alpha/2 =0.025`, and we can use excel, a calculator or a tabel to find the critical value. The excel command would be: "=-NORM.INV(0.025,0,1)".And we see that
`z_(\alpha/2)=1.96`

The standard error is given by the following formula:

`SE=\sqrt{\sigma^2((1)/(n_1)+(1)/(n_2))}`

And replacing we have:

`SE=\sqrt{3^2((1)/(10)+(1)/(10))}=1.342`

Confidence interval

Now we have everything in order to replace into formula (1):

`-3.35-1.96\sqrt{3^2((1)/(10)+(1)/(10))}=-5.980`

`-3.35+1.96\sqrt{3^2((1)/(10)+(1)/(10))}=-0.720`

So on this case the 95% confidence interval would be given by
`-5.980 \leq \mu_1 -\mu_2 \leq -0.720`

If the system of hypothesis are given by:

Null Hypothesis:
`\mu_1 -\mu_2=0`

Alternative hypothesis:
`\mu_1 -\mu_2 \\eq 0`

The statistic would be:

`z=\frac{\bar X_1 -\bar X_2 -0}{\sqrt{\sigma^2((1)/(n_1)+(1)/(n_2))}}`

And if we replace we got:

`z=\frac{65.2 -68.55 }{\sqrt{3^2((1)/(10)+(1)/(10))}}=-2.497`

And the p value would be given by:

`p_v =2*P(Z<-2.497)=0.012`

And with 5% of significance we have enough evidence to reject the null hypothesis since the
`p_v < \alpha`