Derby Leicester is a city planner preparing for a meeting with the mayor. He would like to show that the population mean age of the houses on Lincoln Street is less than the population mean age of the houses on Maple Street so that more resources are allotted to repair Maple Street. Derby uses a previous study to assume that the population standard deviation for the ages of the houses is 7.72 years for Lincoln Street and 8.39 years for Maple Street. Due to limited time, Derby randomly selects a sample of houses on Lincoln Street and the houses on Maple Street from the city’s property records and then records the age of each house in years. The results of the samples are shown in the table below. Explain whether a hypothesis test for the difference between two means of independent samples is appropriate, and if so, determine the null and alternative hypotheses for this hypothesis test, where μ1 is the population mean age of the homes on Lincoln Street and μ2 is the population mean age of the homes on Maple Street.Lincoln StreetMaple Streetx¯¯¯1=50.91 yearsx¯¯¯2=59.27 yearsn1=41n2=37 Select the correct answer below: A) All of the conditions to conduct the hypothesis test are met. The null and alternative hypotheses are {H0:μ1−μ2=0Ha:μ1−μ2>0.B) All of the conditions to conduct the hypothesis test have been met. The null and alternative hypotheses are {H0:μ1−μ2=0Ha:μ1−μ2≠0.C) All of the conditions to conduct the hypothesis test are met. The null and alternative hypotheses are {H0:μ1−μ2=0Ha:μ1−μ2<0.D) Although the population standard deviations are known and the sample sizes are sufficiently large, the samples are not randomly selected and are not independent of each other.E) Although the population standard deviations are known and the samples are randomly selected and independent, the sample sizes are not sufficiently large.F) Although the samples are randomly selected, independent, and sufficiently large, the population standard deviation is not known for both neighborhoods.

Question
Asked Jun 16, 2019

Derby Leicester is a city planner preparing for a meeting with the mayor. He would like to show that the population mean age of the houses on Lincoln Street is less than the population mean age of the houses on Maple Street so that more resources are allotted to repair Maple Street. Derby uses a previous study to assume that the population standard deviation for the ages of the houses is 7.72 years for Lincoln Street and 8.39 years for Maple Street. Due to limited time, Derby randomly selects a sample of houses on Lincoln Street and the houses on Maple Street from the city’s property records and then records the age of each house in years. The results of the samples are shown in the table below. Explain whether a hypothesis test for the difference between two means of independent samples is appropriate, and if so, determine the null and alternative hypotheses for this hypothesis test, where μ1 is the population mean age of the homes on Lincoln Street and μ2 is the population mean age of the homes on Maple Street.

Lincoln Street Maple Street
x¯¯¯1=50.91 years x¯¯¯2=59.27 years
n1=41 n2=37
 

Select the correct answer below: 

A) All of the conditions to conduct the hypothesis test are met. The null and alternative hypotheses are {H0:μ1−μ2=0Ha:μ1−μ2>0.

B) All of the conditions to conduct the hypothesis test have been met. The null and alternative hypotheses are {H0:μ1−μ2=0Ha:μ1−μ2≠0.

C) All of the conditions to conduct the hypothesis test are met. The null and alternative hypotheses are {H0:μ1−μ2=0Ha:μ1−μ2<0.

D) Although the population standard deviations are known and the sample sizes are sufficiently large, the samples are not randomly selected and are not independent of each other.

E) Although the population standard deviations are known and the samples are randomly selected and independent, the sample sizes are not sufficiently large.

F) Although the samples are randomly selected, independent, and sufficiently large, the population standard deviation is not known for both neighborhoods.

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Expert Answer

Step 1

Assumptions:

The assumptions for a hypothesis test of the difference between two population means are:

  • The samples are selected randomly and are independent of each other.
  • The samples are drawn from a normally distributed population.
Step 2

Assumptions for this problem:

The researcher likes to show that the population mean age of the houses on Lincoln street is less than the population mean age of houses on Maple street.

Here, µ1 represents the population mean age of the houses on Lincoln street and µ2 represents the population mean age of the houses on Maple street.

Let σ1 represent the population standard deviation for the ages of the houses in Lincoln street. It is given that σ1 = 7.72 years.

Also, let σ2 represent the population standard deviation for the ages of the houses in Maple street. It is given that σ2 = 8.39 years.

The researcher selects a random sample of houses from Lincoln street and Maple street and the ages of houses in years are recorded.

Thus, the samples are selected randomly and are independent of each other.

It can be assumed that the samples are drawn from a normally distributed population. It was found that for a sample of 41 houses (n1), the sample mean (x1-bar) was 50.91 years in Lincoln street and for a sample of 37 houses (n2), the sample mean (x2-bar) was 59.27 years in Maple street.

Here, the sample sizes are n1 (=41) ≥ 30 and n2 (=37) ≥30.

Since, the population standard deviations are known, and the sample sizes are large (i.e. n1≥30 and n2≥30), the two sample z-test for independence is appropriate here.

Hence, all the conditions to conduct the hypothesis test are met.

Step 3

Determining null and alternative hypothesis for the test:

Null and alternative hypotheses:

Null hypothesis:

Null hypothesis is a statement which is tested for statistical significance in the test. The decision criterion indicates whether the null hypothesis will be rejected or not in the favor of alternative hypothesis. In other words, it can be said that, the null hypothesis is a s...

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