Question

A research council wants to estimate the mean length of time (in minutes) that the average U.S. adult spends watching television using digital video recorders (DVR’s) each day. To determine the estimate, the research council takes random samples of 35 U.S. adults and obtains the following times in minutes.

24	27	26	29	33
21	18	24	23	34
17	15	19	23	25
29	36	19	18	22
16	45	32	12	24
35	14	40	30	19
14	28	32	15	39

From past studies, the research council has found that the standard deviation time is 4.3 minutes and that the population of times is normally distributed.

Construct a 90% confidence interval for the population mean.

Construct a 99% confidence interval for the population mean.

Interpret the results and compare the widths of the confidence intervals.

Test the claim that the mean time spent watching DVR’s is not 20 minutes each day using a significance level of 0.05. Remember that the alternative hypothesis reflect what the researcher's claim. Be sure to state the null and alternative hypotheses.

You may use Stat Disk, TI-84 calculator, or CrunchIt to find the confidence intervals. Be sure to show your results in your post.

Answer 1

The mean of the given data is calculated using excel tool as =AVERAGE(A1:A35)

Mean, M=25.057

a) The formula for estimation is:

μ = M ± Z(s_M)

where:

M = sample mean
Z = Z statistic determined by confidence level
s_M = standard error = √(s²/n)

M = 25.057
Z = 1.64
s_M = √(4.3²/35) = 0.73

μ = M ± Z(s_M)
μ = 25.057 ± 1.64*0.73
μ = 25.057 ± 1.19553

90% CI [23.86147, 26.25253].

b) and 99 % confidence interval as

M = 25.057
Z = 2.58
s_M = √(4.3²/35) = 0.73

μ = M ± Z(s_M)
μ = 25.057 ± 2.58*0.73
μ = 25.057 ± 1.8722

99% CI [23.1848, 26.9292].

c) The Hypotheses are:

Rejection region:

Reject Ho if |Z| > Z _0.025 =1.96

Test Statistic:

P-value:

P-value computed using table shown below as

P-value=0

Conclusion:

Since the P-value << 0.05 ( level of significance ) and |Z| >> Z_0.025 =1.96 hence we reject the null hypothesis and support the claim that the ,mean is different from 20.

Z table