Question

Question 2 (a)A new engineering software is being developed and claim that it is more than 98% compatible with the existing software. A sample of 200 engineering firms tested the new software and 180 of them agreed with the manufacturer. Test at 5% significance level whether the claim is true. (9 marks) (b) The noise level of a newly designed low noise transistor is normally distributed with mean and standard deviation 0.7 dB. The null hypothesis u-24dB is to be tested against μ<24 using a random sample of size 26 transistors. It is decided that the null hypothesis will be rejected if the sample mean is greater than 2.6. Calculate the probability of making i) Type I error, (5 marks) i) Type II error, when in fact μ-2 1 dB (5 marks) A machine is set to produce discs with mean diameter 14 mm. A random sample of 6 discs from a day's production yielded the following diameters, in mm. (c) 13.1 15.0 14.4 13.4 16.1 17.4 Construct a 95% confidence interval for the mean diameter (5 marks) (TOTAL: 24 Marks)

Answer 1

Solution Q2

Part (a)

Let p = population proportion of compatibility.

Claim :

New software is more than 98% (0.98) compatible with the existing software.

We will test this claim using Confidence Interval for p.

Concept Base

Application of confidence interval to take decision on the hypothesis

For any population parameter θ, Null hypothesis: H₀: θ = θ₀ Vs Alternative: H₁: θ ≠ θ₀ is rejected at significance level, α%, if θ₀ is not contained in the 100(1 – α)% Confidence Interval for θ.

Now to work out the solution

100(1 - α) % Confidence Interval for the population proportion, p is: p_cap ± MoE,

where

MoE = Z_α/2[√{p_cap(1 –p_cap)/n}]

with

Z_α/2 is the upper (α/2)% point of N(0, 1),

pcap = sample proportion, and

n = sample size.

Given 5% significance level, α = 0.05.

So, 95% confidence interval for p is: [0.86 < p < 0.94]

Since this interval does not hold 0.98 or more, the claim is not valid. Answer 1

Details of calculations

n	200
X	180
p' = pcap	0.9
F = p'(1-p')/n	0.00045
sqrtF	0.021213
α	0.05
1 - (α/2)	0.975
Zα/2	1.959964
LB	0.858423
UB	0.941577

Part (b)

Concept Base

α = P(Type I Error) = probability of rejecting a null hypothesis when it is true

β = P(Type II Error) = probability of accepting a null hypothesis when it is not true, i.e., Alternative is true.

Now to work out the solution

P(Type I Error)

= P(Xbar > 2.6/given µ = 2.4, σ = 0.7)

= P[Z > {(√26)(2.6 – 2.4)/0.7}] [If X has mean µ, standard deviation σ, and Xbar = sample mean based a sample size n, then Z = {(√n)(Xbar – µ)/σ} ~ N(0, 1)]

= P(Z > 1.4569)

= 0.0726 [Using Excel Function: Statistical NORMSDIST] Answer 2

P(Type II Error)

= P(Xbar < 2.6/given µ = 2.1, σ = 0.7)

= P[Z < {(√26)(2.6 – 2.1)/0.7}] [If X has mean µ, standard deviation σ, and Xbar = sample mean based a

                                                      sample size n, then Z = {(√n)(Xbar – µ)/σ} ~ N(0, 1)]

= P(Z < 3.6422)

= 0.9999 [Using Excel Function: Statistical NORMSDIST] Answer 3

Part (c)

Concept Base

100(1 - α) % Confidence Interval for population mean μ, when σ is not known is: Xbar ± MoE

Where

MoE = (t_{n- 1, α /2})s/√n

with

Xbar = sample mean,

t_{n – 1, α /2} = upper (α/2)% point of t-distribution with (n - 1) degrees of freedom,

s = sample standard deviation and

n = sample size.

Now to work out the solution

Data

i	xi
1	13.1
2	15.0
3	14.4
4	13.4
5	16.1
6	17.4

Summary of calculation

n	6
Xbar	14.9
s	1.639512
√n	2.44949
α	0.05
n - 1	5
tα/2	2.570582
MoE = (s/√n)(tα/2)	1.720562
Lower Bound	13.17944
Upper Bound	16.62056

So, 95% confidence interval for mean diameter is: [13.179 mm < Mean Diameter < 16.621 mm] Answer 4

DONE