Confidence Interval for mean known or unknown
What are the 2 conditions for constructing a Confidence interval?
+/- 1.7507
Decide if this problem is a Type I Error, a Type II Error, or no error
Type II Error
A test is made of H0: μ = 5 versus Ha: μ < 5. The true value of μ is 5, and the null is rejected.
Is this a Type I error, Type II error, or a correct decision?
Type I
A hypothesis test with significance level .05 has a -value . Should we reject the null hypothesis? Explain
Reject the null
A survey of workers in a certain state asked 130 people how long they had been employed in their present job. The sample mean was 6.45 years with a standard deviation of 4.32 years. Construct a 95% confidence interval
df = 130 – 1= 129 CL = 95%, so 𝒕∗ at 129 df for 95% confidence requires the next lowest df, since
129 df is not on the table. Therefore, use 100 df at 95% confidence to get t* = 1.984
6.45 ± 1.984 ( 𝟒.𝟑𝟐
√𝟏𝟑𝟎) = 6.45 ± 0.7517 = (5.698, 7.202)
A recent study indicated that 29% of the 100 women over age 55 in the study were widows.
Find a 90% confidence interval for the true proportion of women over age 55 who are widows.
(0.2154, 0.3646)
Decide if this problem is a Type I Error, a Type II Error, or no error
A newborn baby’s true weight is 7.8 oz. The baby's dad rejects H0
The mean test score on the District wide mathematics test was 62 with a standard deviation of 14. The
math department chair at Barbieland High believes the mean is less than this at her school. She takes a SRS of 16 students and finds a sample mean of 60. Test her claim at the .05 significance level.
The sample does not provide enough evidence to conclude that students at Barbieland scored lower than the district
The city management company claims that 75% of all low income housing is 1500 sq. ft. The tenants
believe the proportion of housing this size is smaller than the claim, and hire an independent engineering firm to test an appropriate hypothesis. He finds that among a random sample of 230 public housing units, 172 of them are 1500 sq. ft.
Is the tenants claim substantiated by this report? Test at the .01 level of significance.
The tenant’s claim that the proportion of low-income housing is less than 75% is not supported by this sample
Construct a 98% confidence interval for the population mean if the sample size is 32, the sample mean is 18.2, and the sample deviation is 2.6.
df = 32 – 1= 31 CL = 98%, so 𝒕∗ at 31 df for 98% confidence = 2.453
18.2 ± 2.453 ( 𝟐.𝟔/ √𝟑𝟐) = 18.2 ± 1.1274 = (17.073, 19.327)
A CBS News/New York Times poll found that 329 out of 763 adults said they would travel to outer space in their lifetime, given the chance. Estimate the true proportion of adults who would like to travel to outer space with 92% confidence.
(0.3998, 0.4626)
Would this be left-tailed, right-tailed, or two-tailed?
Right tailed
The volume of soda in quart soda bottles can reportedly be described by a Normal model with a mean
of 32.3 oz and a standard deviation of 1.2 oz. A consumer advocate group thinks that the volume is actually less than
that. They take a SRS of 50 bottles and find a sample mean of 32.1 oz. Test the claim of the advocate group at the .05
significance level.
The sample did not provide enough evidence to support the
consumer advocate group’s claim that the volume of soda in quart soda bottles is less than the reported 32.2 ounces.
A test is made of H0: μ = 63 versus Ha: μ > 63. The true value of μ is 75, and the null is not rejected.
Is this a Type I error, Type II error, or a correct decision?
Type II
Find the critical t value needed to construct a confidence interval of the given level with the given sample size. Round answers to three decimal places.
For level 90% and sample size 25 → df = n – 1 = 24, so t* = 1.711
You play a game and win 136 out of 270 times.
Find a 95% confidence interval for the probability of winning the game.
(0.4441, 0.5633)
State an appropriate null and alternative hypothesis
H0: μ = 83.49 H1: μ (does not equal) =/49
A government report indicates that in 2012 the mean cost for a normal delivery of a baby was $9,775. A
consumer group believes that one particular hospital group has lower mean for baby delivery costs. The take a random sample of the costs for 40 deliveries in the past year with this particular hospital group and finds a mean cost of $9,700 with a standard deviation of $325. Conduct a hypothesis test at the .05 significance level to test the claim of the consumer group.
The sample does not provide enough evidence to support the consumer group’s claim that the mean baby delivery cost is lower for that particular hospital
A state university wants to increase its retention rate of 4% for college graduates entering graduate
school. After implementing several new programs during the last two years, the university evaluates its records and
finds that among 5250 graduates, 246 enter graduate school. Did the new programs improve retention for graduate
school? Test at the .05 level of significance.
The sample provided enough evidence to conclude that the new
programs improved retention for graduate school at this university
A sample consists of 75 TV sets purchased several years ago. The replacement times of those TV sets have a mean of 8.2 years. Assume σ= 1.1 years. n = 75 𝒙̅ = 𝟖. 𝟐 𝝈 = 𝟏. 𝟏
Construct the 90% Confidence Interval
90%: 8.2 ± 1.645 ( 𝟏.𝟏/√𝟕𝟓) = 8.2 ± 0.2089 = (7.991, 8.409)
An economist wants to estimate the mean income for the first year of work for a college graduate who has had the profound wisdom to take a statistics course. How many such incomes must be found if we want to find a 95% confidence interval within $500 of the sample mean? Assume a standard deviation of $6250.
n = ((𝒛∗(𝛔)/𝐦.𝐞.) )𝟐 = ((𝟏.𝟗𝟔(𝟔𝟐𝟓𝟎))/5𝟎𝟎 )𝟐 = 600.3 → need a sample of 601 incomes
State an appropriate null and alternative hypothesis
A newborn baby was weighed at 7.8 oz. The baby's dad believes that the baby is heavier.
H0: μ = 7.8 H1: μ > 7.8
The average number of days absent per student per year at West Valley School District is 17 days with a
standard deviation of 4 days. The counselor believes that the mean is greater than that for Freshmen. She takes a SRS
of 25 Freshmen and finds a sample mean of 19 days. Test her claim at the .05 significance level.
This sample supports the claim that the mean number of absent
days for West Valley School District freshman is greater than the mean for the whole district.
A new manager, hired at a large warehouse, was told to reduce the 26% employee sick leave. The
manager introduced a new incentive program for employees with perfect attendance. He evaluated the program after 6
months and found that out of 245 employees, 51 had been out sick. Has the proportion of employees taking sick leave
decreased significantly? Test at a .05
There is enough evidence for the new manager to conclude that the
proportion of employees taking sick leave has decreased significantly.