Question

School is interested in knowing the average height of undergraduate students but do not have time to measure all the students. 100 students were randomly selected and it is found that the average height of these 100 students is 1.60 metres with a standard deviation of 0.3 metres.

a) State the point estimate of the average height of undergraduate students.

b) Construct an interval estimate of the average height of students with 99% confidence.

c) The respond rate of the above survey was 85%, i.e. 85% of the students contacted were willing to participate in the survey. Construct an interval estimate of the proportion of students population who are willing to participate in other similar survey with 95% confidence.

d) The Student Services Centre is not very convinced of the result and would like to conduct a second survey. It would like to estimate the mean population height of students to be within 0.05 metres and be 99% confident, assuming the population standard deviation is 0.3 metres, how large a sample is necessary to achieve the accuracy stated?

Answer 1

Here we have : n = 100, = 1.60, s = 0.3

a) Point estimate of the average height of undergraduate student is sample mean .

i.e. = 1.60

b) Here sample size is greater than 30. Hence we construct z interval.

The 99% confidene interval for estimate of the average height of students is given by,

( - E , + E )

Where,

c = 0.99, = 1-c = 1-0.99 = 0.01,

Zc = ------------( using excel formula "=norm.s.inv(0.995)" )

Hence the margin of error is given by,

= 0.08

Hence the 99% confidence interval is,'

( 1.60 - 0.08, 1.60 + 0.08 )

( 1.52 , 1.68 )

c) The respond rate of survey = p = 0.85

We need to find, 95% confidence interval for the proportion of students who are willing to participate is given by,

( p -E , p + E )

Where,

c =0.95, = 1- c = 1-0.95 = 0.05

--------( using excel formula " =norm.s.inv(0.05)" )
Hence the margin of error is given by,

= 0.07

Hence 95% confidence interval is given by,

( 0.85 - 0.07 , 0.85 + 0.07 )

( 0.78 , 0.92 )

d)

The mean population height of students to be within 0.05 metres . This is denoted by E = 0.05.

Population standard deviation = = 0.3

Confidence level is 0.99.

c = 0.99, = 1-c = 1-0.99 = 0.01,

Zc = ------------( using excel formula "=norm.s.inv(0.995)" )

Hence, required sample size is given by,

  

= 239.63

   240