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Fuel efficiency of manual and automatic cars: Each year the US Environmental Protection Agency (EPA) releases fuel economy data on cars manufactured in that year. Below are summary statistics on fuel efficiency (in miles/gallon) from random samples of cars with manual and automatic transmissions manufactured in 2012. Do these data provide strong evidence of a difference between the average fuel efficiency of cars with manual and automatic transmissions in terms of their average city mileage at the 5% level of significane? Assume that conditions for inference are satisfied.

Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)

Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)

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ISBN:9781337111348

Author:Bruce Crauder, Benny Evans, Alan Noell

Publisher:Bruce Crauder, Benny Evans, Alan Noell

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Problem 5E: Population Genetics In the study of population genetics, an important measure of inbreeding is the...

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Fuel efficiency of manual and automatic cars: Each year the US Environmental Protection Agency (EPA)
releases fuel economy data on cars manufactured in that year. Below are summary statistics on fuel
efficiency (in miles/gallon) from random samples of cars with manual and automatic transmissions
manufactured in 2012. Do these data provide strong evidence of a difference between the average fuel
efficiency of cars with manual and automatic transmissions in terms of their average city mileage at the 5%
level of significane? Assume that conditions for inference are satisfied.
City MPG, Automatic
City MPG, Manual
Mean
16.12
19.85
SD
3.58
4.51
26
26
35
25 -
15
automatic
manual
City MPG
The hypotheses for this test are:
O Ho: Hautomatic = Pmanual
Ha: Pautomatic * Pmanual
%3D
O Ho: Hautomatic = Pmanual
Ha: Pautomatic < Pmanual
O Ho: Hautomatic = Pmanual
Ha: Hautomatic > Pmanual
The test statistic is:
(please round to two decimal places)
The p-value is:
(please round to four decimal places)

Expert Solution

Step 1

Let $μ_{automatic}$ denote the average city mileage of automatic transmissions, and $μ_{manual}$ denote the average city mileage of manual transmissions. The test is whether there is difference between the average fuel efficiency of cars with manual and automatic transmissions in terms of their average city mileage. The hypothesis is,

Null hypothesis:

$H_{0} : μ_{automatic} = μ_{manual}$

Alternative hypothesis:

$H_{a} : μ_{automatic} \neq μ_{manual}$

The pooled variance is,

$\begin{array}{rcl} s_{p}^{2} & = & \frac{(n_{1} - 1) s_{1}^{2} + (n_{2} - 1) s_{2}^{2}}{n_{1} + n_{2} - 2} \\ = & \frac{(26 - 1) 3 . 58^{2} + (26 - 1) 4 . 51^{2}}{26 + 26 - 2} \\ = & \frac{828.9125}{50} \\ = & 16.57825 \end{array}$

The pooled variance is 16.57825.

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Urban Travel Times Population of cities and driving times are related, as shown in the accompanying table, which shows the 1960 population N, in thousands, for several cities, together with the average time T, in minutes, sent by residents driving to work. City Population N Driving time T Los Angeles 6489 16.8 Pittsburgh 1804 12.6 Washington 1808 14.3 Hutchinson 38 6.1 Nashville 347 10.8 Tallahassee 48 7.3 An analysis of these data, along with data from 17 other cities in the United States and Canada, led to a power model of average driving time as a function of population. a Construct a power model of driving time in minutes as a function of population measured in thousands b Is average driving time in Pittsburgh more or less than would be expected from its population? c If you wish to move to a smaller city to reduce your average driving time to work by 25, how much smaller should the city be?
Population Genetics In the study of population genetics, an important measure of inbreeding is the proportion of homozygous genotypesthat is, instances in which the two alleles carried at a particular site on an individuals chromosomes are both the same. For population in which blood-related individual mate, them is a higher than expected frequency of homozygous individuals. Examples of such populations include endangered or rare species, selectively bred breeds, and isolated populations. in general. the frequency of homozygous children from mating of blood-related parents is greater than that for children from unrelated parents Measured over a large number of generations, the proportion of heterozygous genotypesthat is, nonhomozygous genotypeschanges by a constant factor 1 from generation to generation. The factor 1 is a number between 0 and 1. If 1=0.75, for example then the proportion of heterozygous individuals in the population decreases by 25 in each generation In this case, after 10 generations, the proportion of heterozygous individuals in the population decreases by 94.37, since 0.7510=0.0563, or 5.63. In other words, 94.37 of the population is homozygous. For specific types of matings, the proportion of heterozygous genotypes can be related to that of previous generations and is found from an equation. For mating between siblings 1 can be determined as the largest value of for which 2=12+14. This equation comes from carefully accounting for the genotypes for the present generation the 2 term in terms of those previous two generations represented by for the parents generation and by the constant term of the grandparents generation. a Find both solutions to the quadratic equation above and identify which is 1 use a horizontal span of 1 to 1 in this exercise and the following exercise. b After 5 generations, what proportion of the population will be homozygous? c After 20 generations, what proportion of the population will be homozygous?

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