Question

In this assignment students will demonstrate their understanding of the distribution of means doing all steps of hypothesis testing.
For each problem students will write out all steps of hypothesis testing including populations, hypotheses, cutoff scores, and all relevant calculations. Assignments will be typed and uploaded in a word document to blackboard.

1.A nationwide survey in 1995 revealed that U.S. grade-school children spend an average of µ = 8.4 hours per week doing homework. The distribution is normal with σ = 3.2. Last year, a sample of n = 100 grade-school children was given the same survey. For this sample, the mean number of homework hours was 7.1. Has there been a significant change in the homework habits of grade-school children? Test with α = .05.

2.On the basis of her newly developed technique, a student believes she can reduce the amount of time schizophrenics spend in an institution. As director of training at a nearby institution, you agree to let her try her method on 20 schizophrenics, randomly sampled from your institution. The mean duration that schizophrenics stay at your institution is 85 weeks, with a standard deviation of 15 weeks. The scores are normally distributed. The results of the experiment show that patients treated by the student stay at the institution a mean duration of 78 weeks. What do you conclude about the student’s technique? Use α = .05.

3.A psychologist has developed a standardized test for measuring the vocabulary skills of 4-year-old children. The scores on the test form a normal distribution with μ = 60 and σ = 10. A researcher would like to use this test to investigate the idea that children who grow up with no siblings develop vocabulary skills at a different rate than children in large families. A sample of n = 25 children is obtained, and the mean test score for this sample is 63. On the basis of this sample, can the researcher conclude that vocabulary skills for children with no siblings are significantly different from those of the general population? Test at the .01 level of significance.

4.The average age for licensed drivers in a county is 42.6, with a standard deviation of 12, and the distribution is approximately normal. A county police officer was interested in whether the average age of those receiving speeding tickets is less that the average age of the population who has a license. She obtained a sample of 16 drivers with speeding tickets. The average age for this sample was 34.4. Do all the steps of hypothesis testing using the 0.01 significance level.

Answer 1

1.

Population: US Grade School children

H0: = 8.4

H1: 8.4

XN(8.4,3.22)

=3.2, n=100, =7.1

N(8.4,3.22/100)

Test statistic, Z=(-)/(/) = (7.1-8.4)/(3.2/10) = - 1.3/.32 = - 4.06

Zcrit = Z/2=0.025 = 1.96<|Z|

So , reject null hypothesis. So, There has been a significant change in home work habit of grade school children.

2.

Population: Schizophrenics of my school

H0: = 85

H1: <85

XN(85,152)

=15, n=20, =78

N(85,152/20)

Test statistic, Z=(-)/(/) = (78-85)/(15/) = - 7/3.35 = - 2.09

Zcrit = -Z=0.05 = -1.64>Z or |Zcrit|<|Z|

So , reject null hypothesis. So, Students technique has reduced the amount of time Schizophrenics spend in an institution.

3.

Population: 4 year old children

H0: = 60

H1: 60

XN(60,102)

=10, n=25, =63

N(60,102/25)

Test statistic, Z=(-)/(/) = (63-60)/(10/5) = 3/2 = 1.5

Zcrit = Z/2=0.005 = 2.58>|Z|

So , fail to reject null hypothesis. So, researcher cannot conclude that vocabulary skill of children with with no siblings are significantly different from those of the general population.

4.

Population: licensed drivers in a county

H0: = 42.6

H1: <42.6

XN(42.6,122)

=12, n=16, =34.4

N(42.6,122/15)

Test statistic, Z=(-)/(/) = (34.4-42.6)/(12/) = - 8.2/3 = - 2.73

Zcrit = -Z=0.01 = -2.33>Z or |Zcrit|<|Z|

So , reject null hypothesis. So, the average age of those receiving speeding tickets is less than the average age of the population who has a license.