+(d) (d) Construct a hypothesis test about whether the true mean is 70, use that

Question

+(d) (d) Construct a hypothesis test about whether the true mean is 70, use that significance level that is equivalent to part (b).

A random sample of size 16 is drawn from a normal distribution with HE 70 and o E 5. The mean of the sample is 68.45 and s 2.73 (a) Calculate a 90% z-interval for Au assuming that you know a 3. (b) Calculate a 90% t-interval for Au assuming that you do not know or (c) Which interval is shorter for this sample? Which interval would be shorter on the average if a large number of samples are drawn from this normal distribution, and z and t intervals are calculated for each sample? Explain.

Explanation / Answer

a) for std error =std deviation/(n)1/2 =3/(16)1/2 =0.75

for 90% CI, z=1.6449

hence confidence interval =sample mean -/+z*std error =67.216 ; 69.684

b) here as we do not know population std deviaiton we will use t distribution

std error =2.73/(16)1/2 =0.6825

for 15 degree of freedom and 90% CI, t=2.1314

hence confidence interval =sample mean -/+t*std error =67.216 ; 66.9953 ; 69.9047

c)z interval is shorter and it will be z only for larger number of sample size to be shorter as t distribution is always wider then z distribution to compensate for unknown std deviation

d) as 70 does not lies in our confidence level as true value of mean , we reject null that mean is 70

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