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STATS 151 > Chpt 9 - Hypothesis Tests > Flashcards

Chpt 9 - Hypothesis Tests Flashcards

1
Q

What is a hypothesis?

A

An educated guess about something in the world

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2
Q

What is hypothesis testing?

A

An important statistical procedure to test a hypothesis. It evaluates the null hypothesis and alternative hypothesis and determines which statement is best supported by the experiment results

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3
Q

What are the two hypothesis set up for hypothesis testing?

A

Ho: null hypothesis

Ha: alternative hypothesis

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4
Q

Which hypothesis is the outcome you want to prove to be true?

A

Ha: alternative hypothesis

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5
Q

Why is the null hypothesis assumed to be true?

A

We cannot prove that anything is actually true, we can only prove a statement is not true. So we test the null hypothesis because we can say it is not true and reject it or that it is “true enough” that it cannot be rejected

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6
Q

What happens that results in a type 1 hypothesis error?

How is a type 1 error notated?

A

The null hypothesis is true but we rejected it

noted as α (alpha)

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7
Q

What happens that results in a type 2 hypothesis error?

How is a type 2 error notated?

A

The null hypothesis is false but it is not rejected

Notated as β

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8
Q

What type of error results when a null hypothesis is true but is rejected?

A

Type 1 α error

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9
Q

What type of error results when a null hypothesis is true but is not rejected?

A

Nothing, this is what should happen

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10
Q

What type of error results when a null hypothesis is false but it is rejected?

A

Nothing, this is what should happen

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11
Q

What type of error results when a null hypothesis is false but it is not rejected?

A

Type 2 β error

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12
Q

If a machine is working properly, the mean volume should be 100 mL. Set up the hypotheses to test if the machine is not working correctly

A

Ho: μ = 100

Ha: μ ≠ 100

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13
Q

What is the probability of making a type 1 error called?

A

Significance level α

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14
Q

What is the probability of making a type 2 error called?

A

Type 2 error probability β

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15
Q

If the sample size is fixed, the smaller the significance level α, the ______ the type II error probability β and vice versa

A

larger

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16
Q

In a hypothesis test, because type II error probability β is not easy to define, what do we usually control instead?

A

significance level α

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17
Q

There are 2 methods of hypothesis testing for one population mean and which one we chose depends on if the population standard deviation is known. What are the 2 methods?

A

Z test when we know the population standard deviation

t test when we do NOT know the population standard deviation

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18
Q

What is the hypothesized mean equal to and why?

A

μ = μo

because the Ho is assumed to be true

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19
Q

What is the statistic that measures the distance between the sample mean x̄ and the hypothesized mean μo in one population mean hypothesis test called?

A

Test statistic

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20
Q

What is the relationship between the significance level and the type II error probability?

A

As the significance level decreases, the type II error probability increases

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21
Q

If the true population of mean μ of X is equal to μo and the population standard deviation is σ, what is the mean and standard deviation of x̄?

A

Mean = μo
Standard deviation = σ/√n

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22
Q

When sample size n is large enough or the parent distribution is normal, what distribution does the sample mean follow?

A

approximately normal distribution

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23
Q

What is the test statistic used for hypothesis testing when σ is known? What does this mean for the mean and standard deviation?

A

σ/√n

Mean of 0, standard deviation of 1

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24
Q

What is the critical value approach to perform hypothesis tests?

A

It finds a threshold, or critical value, and compares that to the test statistic. It can be two-tailed, left-tailed, and right-tailed depending on the alternative hypothesis

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25
Q

What type of hypothesis test is:

Ho: μo = 0
Ha: μo ≠ 0

How many critical values exist in this case?

When do we reject Ho?

What is the value of α?

What is the critical value?

A

two-tailed hypothesis

There are two critical values, C1 on the low end and C1 on the upper end.

We reject Ho if they are smaller than C1 or larger than C2

α is the value under the density curve less than C1 and more than C2

C1 & C2 = ±Zα/2

OR

C1 & C2 = ±tα/2

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26
Q

Describe the rejection region in a two tailed hypothesis and what is α?

A

In a two tailed hypothesis, the area below C1 and above C2 are the rejection regions. their sum is α. The significance level of α is set up prior to doing the test statistic to ensure we set an adequate value rather than make the numbers explain what we want them to.

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27
Q

When we know σ, and the sample size is large enough or parent distribution is normal, the test statistic follows ______distribution and C1 is _____ and C2 is _______.

A

N(0,1)

C1: -Zα/2

C2: Zα/2

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28
Q

A machine fills lotion bottles whose mean volume is expected to be 100 mL. Exact volumes of 50 bottles were measured. The sample mean of this sample is 98 mL and assume that the population standard deviation is 5 mL. A researcher claims that the machine is NOT working properly. We want to test the statement is correct at the 5% significance level.

What are the hypotheses?

A

Ho: μ = 100

Ha: μ ≠ 100

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29
Q

A machine fills lotion bottles whose mean volume is expected to be 100 mL. Exact volumes of 50 bottles were measured. The sample mean of this sample is 98 mL and assume that the population standard deviation is 5 mL. A researcher claims that the machine is NOT working properly. We want to test the statement is correct at the 5% significance level.

Check assumptions

A

Normal distribution, in this case we don’t know the parent population distribution, but we do have a sample size of 50 which is large enough to be an approximately normal distribution

Test is a simple random sample

σ is known

30
Q

A machine fills lotion bottles whose mean volume is expected to be 100 mL. Exact volumes of 50 bottles were measured. The sample mean of this sample is 98 mL and assume that the population standard deviation is 5 mL. A researcher claims that the machine is NOT working properly. We want to test the statement is correct at the 5% significance level.

Find the critical values

A

Since the significance value is

α = 5% = 0.05

This is a two-tailed hypothesis so:

α/2 = 0.05/2 = 0.025

C1 = -Z0.025 = -1.96

C2 = Z0.025 = 1.96

so the rejection regions are below -1.96 and above 1.96

31
Q

A machine fills lotion bottles whose mean volume is expected to be 100 mL. Exact volumes of 50 bottles were measured. The sample mean of this sample is 98 mL and assume that the population standard deviation is 5 mL. A researcher claims that the machine is NOT working properly. We want to test the statement is correct at the 5% significance level.

Calculate the test statistic

A

Zo =

x̄-μo 98-100
——- = ———- = -2.83
σ/√n 5/√50

32
Q

A machine fills lotion bottles whose mean volume is expected to be 100 mL. Exact volumes of 50 bottles were measured. The sample mean of this sample is 98 mL and assume that the population standard deviation is 5 mL. A researcher claims that the machine is NOT working properly. We want to test the statement is correct at the 5% significance level.

Compare:

C1: -1.96
C2: 1.96
Test: -2.83

A

Since Zo is -2.83 it falls in the rejection region; therefore, we reject Ho: μ = 100

33
Q

A machine fills lotion bottles whose mean volume is expected to be 100 mL. Exact volumes of 50 bottles were measured. The sample mean of this sample is 98 mL and assume that the population standard deviation is 5 mL. A researcher claims that the machine is NOT working properly. We want to test the statement is correct at the 5% significance level.

We rejected Ho: μ = 100; so interpret the results in context

A

At the 5% significance level, the data provides sufficient evidence to conclude that the mean volume is not 100 mL

34
Q

What type of hypothesis test is:

Ho: μo = > 0
Ha: μo < 0

How many critical values exist in this case?

When do we reject Ho?

What is the value of α?

What is the critical value?

A

Left tailed hypothesis

1 critical value C1

We reject Ho when the test statistic value is smaller than C1 (so the numbers smaller than C1 are the rejection region)

α is the value under the density curve less than C1

C1 = -Zα or -tα

35
Q

A researcher claims that the average weight of 5 year old girls is less than 24kg. A simple random sample of 36 of 5 year old girls picked and the weight of each girl is measured. The average weight of the sample is 20kg. Assume that the population standard deviation is 2kg. We are going to use α = 0.01

What are the hypotheses?

A

Ho: μ => 24

Ha: μ < 24

36
Q

A researcher claims that the average weight of 5 year old girls is less than 24kg. A simple random sample of 36 of 5 year old girls picked and the weight of each girl is measured. The average weight of the sample is 20kg. Assume that the population standard deviation is 2kg. We are going to use α = 0.01

Check the assumptions

A

Normal distribution, in this case we don’t know the parent population distribution, but we do have a sample size of 36 which is large enough to be an approximately normal distribution

Test is a simple random sample

σ is known

37
Q

A researcher claims that the average weight of 5 year old girls is less than 24kg. A simple random sample of 36 of 5 year old girls picked and the weight of each girl is measured. The average weight of the sample is 20kg. Assume that the population standard deviation is 2kg. We are going to use α = 0.01

Find the critical values

A

α = 0.01

Only 1 critical value in a left sided test so

C1 = -Z0.01 = -2.33

So the rejection region includes values smaller than -2.33

38
Q

A researcher claims that the average weight of 5 year old girls is less than 24kg. A simple random sample of 36 of 5 year old girls picked and the weight of each girl is measured. The average weight of the sample is 20kg. Assume that the population standard deviation is 2kg. We are going to use α = 0.01

Calculate the test statistic

A

Zo =

x̄-μo 20-24
——- = ———- = -12
σ/√n 2/√36

39
Q

A researcher claims that the average weight of 5 year old girls is less than 24kg. A simple random sample of 36 of 5 year old girls picked and the weight of each girl is measured. The average weight of the sample is 20kg. Assume that the population standard deviation is 2kg. We are going to use α = 0.01

Compare:
C1: -2.33
Test statistic: -12

A

Since Zo -12 falls into the rejection region, we reject Ho

40
Q

A researcher claims that the average weight of 5 year old girls is less than 24kg. A simple random sample of 36 of 5 year old girls picked and the weight of each girl is measured. The average weight of the sample is 20kg. Assume that the population standard deviation is 2kg. We are going to use α = 0.01

Interpret:
Since Zo -12 falls into the rejection region, we reject Ho

A

At the 1% significance level, the data provides sufficient evidence to conclude that the average weight of 5 year old girls is less than 24 kg

41
Q

What type of hypothesis test is:

Ho: μo = < 0
Ha: μo > 0

How many critical values exist in this case?

When do we reject Ho?

What is the value of α?

What is the critical value?

A

Right tailed hypothesis

One, which is C2

We reject Ho when the test statistic is greater than C2 (which is our rejection region)

α is the value under the density curve more than C2

C2 = Zα or tα

42
Q

The average weight of new born babies in 2014 was 7lbs. Determine whether the average weight in 2015 has increased.

A simple random sample of 100 new born babies are picked and the weight of each baby is measured. The average weight of the sample is 7.4 lbs. Assume that the population standard deviation is 2lb. Suppose we are using a significance level of 1%.

Set up the hypotheses

A

Ho: μ = < 7

Ha: μ > 7

43
Q

The average weight of new born babies in 2014 was 7lbs. Determine whether the average weight in 2015 has increased.

A simple random sample of 100 new born babies are picked and the weight of each baby is measured. The average weight of the sample is 7.4 lbs. Assume that the population standard deviation is 2lb. Suppose we are using a significance level of 1%.

Check the assumptions

A

Normal distribution, in this case we don’t know the parent population distribution, but we do have a sample size of 100 which is large enough to be an approximately normal distribution

Test is a simple random sample

σ is known

44
Q

The average weight of new born babies in 2014 was 7lbs. Determine whether the average weight in 2015 has increased.

A simple random sample of 100 new born babies are picked and the weight of each baby is measured. The average weight of the sample is 7.4 lbs. Assume that the population standard deviation is 2lb. Suppose we are using a significance level of 1%.

Find the critical values

A

α = 0.01

Only 1 critical value in a right sided test so

C2 = Z0.01 = 2.33

So the rejection region includes values larger than 2.33

45
Q

The average weight of new born babies in 2014 was 7lbs. Determine whether the average weight in 2015 has increased.

A simple random sample of 100 new born babies are picked and the weight of each baby is measured. The average weight of the sample is 7.4 lbs. Assume that the population standard deviation is 2lb. Suppose we are using a significance level of 1%.

Calculate the test statistic

A

Zo =

x̄-μo 7.4-7
——- = ———- = 2
σ/√n 2/√100

46
Q

The average weight of new born babies in 2014 was 7lbs. Determine whether the average weight in 2015 has increased.

A simple random sample of 100 new born babies are picked and the weight of each baby is measured. The average weight of the sample is 7.4 lbs. Assume that the population standard deviation is 2lb. Suppose we are using a significance level of 1%.

Compare

Test statistic is 2
Critical value is 2.33

A

Since Zo is 2, it is not in the rejection region, therefore we DO NOT reject Ho

47
Q

The average weight of new born babies in 2014 was 7lbs. Determine whether the average weight in 2015 has increased.

A simple random sample of 100 new born babies are picked and the weight of each baby is measured. The average weight of the sample is 7.4 lbs. Assume that the population standard deviation is 2lb. Suppose we are using a significance level of 1%.

Interpret

Since Zo is 2, it is not in the rejection region, therefore we DO NOT reject Ho

A

At the 1% significance level, the data does NOT provide sufficient evidence to conclude that the average weight of new born babies in 2015 is heavier than the new born babies in 2014

48
Q

What is the P value in a two-tailed alternative hypothesis test?

A

It is the probability that the test statistic is greater than the absolute value of the observed test statistic value or smaller than the negative absolute value given that Ho: μ = μo is true

Because the rejection regions on both sides are equal, the P value probability is the sum of both OR two times either value so:

P value = P(Z < -Z0) + P(Z > Zo)
OR
P value = 2P(Z < -Zo) (2 x the left value)
OR
P value = 2P(Z > Zo) (2 x the right value)

This holds true when using to instead of Zo

49
Q

What is the P value in a left-tailed alternative hypothesis test?

A

The P value is the probability that the test statistic is smaller than the observed test statistic value assuming that Ho: μ = μo is true

P value = P(Z < -Zo)

This holds true when using to instead of Zo

50
Q

What is the P value in a right-tailed alternative hypothesis test?

A

The P value is the probability that the test statistic is larger than the observed test statistic value assuming that Ho: μ = μo is true

P value = P(Z > Zo)

This holds true when using to instead of Zo

51
Q

What assumptions need to be checked for a Z test statistic to be used?

A

Simple random sample

σ is known

There is a normally distributed parent population OR a large enough sample (n>30)

52
Q

What hypothesis test should be done if σ is unknown?

What is the formula for this test statistic?

What distribution does it follow?

A

T test

    x̄-μo to = -------
     s/√n

t-distribution with degrees of freedom n-1

53
Q

The average person watched 5.6 hours of television per day in 2015. the average hours of television per day of a SRS (49 people) in 2016 is 4.8 hours and its standard deviation is 3.6 hours. At the 10% significance level, does the data provide sufficient evidence to conclude that the hours of television watched per day in 2016 by the average person differed from that in 2015

Set up the hypotheses

A

Ho: μ = 5.6

Ha: μ ≠ 5.6

54
Q

The average person watched 5.6 hours of television per day in 2015. the average hours of television per day of a SRS (49 people) in 2016 is 4.8 hours and its standard deviation is 3.6 hours. At the 10% significance level, does the data provide sufficient evidence to conclude that the hours of television watched per day in 2016 by the average person differed from that in 2015

Check assumptions

A

Normal distribution, in this case we don’t know the parent population distribution, but we do have a sample size of 49 which is large enough to be an approximately normal distribution

Test is a simple random sample

σ is unknown so we need to do a t-test

55
Q

The average person watched 5.6 hours of television per day in 2015. the average hours of television per day of a SRS (49 people) in 2016 is 4.8 hours and its standard deviation is 3.6 hours. At the 10% significance level, does the data provide sufficient evidence to conclude that the hours of television watched per day in 2016 by the average person differed from that in 2015

Find critical value

A

α = 10% = 0.1

α/2 = 0.05

degrees of freedom = n-1 = 49 -1 = 48

C1 = -tα/2 = -t0.05 = -1.677

C2 = tα/2 = t0.05 = 1.677

56
Q

The average person watched 5.6 hours of television per day in 2015. the average hours of television per day of a SRS (49 people) in 2016 is 4.8 hours and its standard deviation is 3.6 hours. At the 10% significance level, does the data provide sufficient evidence to conclude that the hours of television watched per day in 2016 by the average person differed from that in 2015

Calculate test statistic

A

to =

x̄-μo 4.8-5.6
——- = ———–
s/√n 3.6/√49

= -1.56

57
Q

The average person watched 5.6 hours of television per day in 2015. the average hours of television per day of a SRS (49 people) in 2016 is 4.8 hours and its standard deviation is 3.6 hours. At the 10% significance level, does the data provide sufficient evidence to conclude that the hours of television watched per day in 2016 by the average person differed from that in 2015

Compare:

test statistic is -1.56
C1 = -1.677
C2 = 1.677

A

Since to is -1.56 it is not in the rejection region; therefore we DO NOT reject Ho

58
Q

The average person watched 5.6 hours of television per day in 2015. the average hours of television per day of a SRS (49 people) in 2016 is 4.8 hours and its standard deviation is 3.6 hours. At the 10% significance level, does the data provide sufficient evidence to conclude that the hours of television watched per day in 2016 by the average person differed from that in 2015

Interpret;

Since to is -1.56 it is not in the rejection region; therefore we DO NOT reject Ho

A

At the 10% significance level, the data does not provide sufficient evidence to conclude that the hours of television watched per day in 2016 by the average person differed from that in 2015

59
Q

The manufacturer of a new car, the Orion, claims that a typical car gets 26 miles per gallon. Some consumers claim that the gas mileage is less. A simple random sample of 100 Orion was selected. Their mean gas mileage is 25.9 mpg and the sample standard deviation is 1.4 mpg. At 5% significance level, does the data provide sufficient evidence to support the consumer’s claim?

Set up the hypotheses

A

Ho: μ => 26

Ha: μ < 26

60
Q

The manufacturer of a new car, the Orion, claims that a typical car gets 26 miles per gallon. Some consumers claim that the gas mileage is less. A simple random sample of 100 Orion was selected. Their mean gas mileage is 25.9 mpg and the sample standard deviation is 1.4 mpg. At 5% significance level, does the data provide sufficient evidence to support the consumer’s claim?

Check the assumptions

A

Normal distribution - there is not information about the parent population, but the sample size is large enough

Simple random sample

σ is unknown so we need to use t-test

61
Q

The manufacturer of a new car, the Orion, claims that a typical car gets 26 miles per gallon. Some consumers claim that the gas mileage is less. A simple random sample of 100 Orion was selected. Their mean gas mileage is 25.9 mpg and the sample standard deviation is 1.4 mpg. At 5% significance level, does the data provide sufficient evidence to support the consumer’s claim?

Determine the critical values

A

α = 5% = 0.05

degrees of freedom = n-1 = 100-1 = 99

C1 = -t0.05 = -1.661

62
Q

The manufacturer of a new car, the Orion, claims that a typical car gets 26 miles per gallon. Some consumers claim that the gas mileage is less. A simple random sample of 100 Orion was selected. Their mean gas mileage is 25.9 mpg and the sample standard deviation is 1.4 mpg. At 5% significance level, does the data provide sufficient evidence to support the consumer’s claim?

Calculate the test statistic

A

to =

x̄-μo 25.9-26
——- = ————- = -0.714
s/√n 1.4/√100

63
Q

The manufacturer of a new car, the Orion, claims that a typical car gets 26 miles per gallon. Some consumers claim that the gas mileage is less. A simple random sample of 100 Orion was selected. Their mean gas mileage is 25.9 mpg and the sample standard deviation is 1.4 mpg. At 5% significance level, does the data provide sufficient evidence to support the consumer’s claim?

Compare

Test statistic is -0.714
C1 is -1.661

A

Since to is 0.714 is is not within the rejection region, therefore we do not reject Ho

64
Q

The manufacturer of a new car, the Orion, claims that a typical car gets 26 miles per gallon. Some consumers claim that the gas mileage is less. A simple random sample of 100 Orion was selected. Their mean gas mileage is 25.9 mpg and the sample standard deviation is 1.4 mpg. At 5% significance level, does the data provide sufficient evidence to support the consumer’s claim?

Interpret

Since to is 0.714 is is not within the rejection region, therefore we do not reject Ho

A

At the 5% significance level, the data does not provide sufficient evidence to support the consumers’ claim

65
Q

The supplier claims that the main life time of their light bulbs is more than 15 000 hours. Steve randomly samples 36 bulbs and recorded their life times. The sample mean of the sample is 16 000 hours and the sample standard deviation is 2.8 thousand hours. At the 10% significance level, is there strong enough evidence to support the supplier’s claim?

Set up the hypotheses

A

Ho: μ =< 15

Ha: μ > 15

66
Q

The supplier claims that the main life time of their light bulbs is more than 15 000 hours. Steve randomly samples 36 bulbs and recorded their life times. The sample mean of the sample is 16 000 hours and the sample standard deviation is 2.8 thousand hours. At the 10% significance level, is there strong enough evidence to support the supplier’s claim?

Check the assumptions

A

Normal distribution: there is not parent population information, but the sample is greater than n=30 so it will be approximately normal

simple random sample

σ is unknown, so need a t-test

67
Q

The supplier claims that the main life time of their light bulbs is more than 15 000 hours. Steve randomly samples 36 bulbs and recorded their life times. The sample mean of the sample is 16 000 hours and the sample standard deviation is 2.8 thousand hours. At the 10% significance level, is there strong enough evidence to support the supplier’s claim?

Calculate the critical value

A

α = 10% = 0.1

degrees of freedom = n-1 = 36-1 = 35

C2 = t0.1 = 1.306

68
Q

The supplier claims that the main life time of their light bulbs is more than 15 000 hours. Steve randomly samples 36 bulbs and recorded their life times. The sample mean of the sample is 16 000 hours and the sample standard deviation is 2.8 thousand hours. At the 10% significance level, is there strong enough evidence to support the supplier’s claim?

Calculate the test statistic

A

to =

x̄-μo 16-15
——- = ————- = 2.14
s/√n 2.8/√36

69
Q

The supplier claims that the main life time of their light bulbs is more than 15 000 hours. Steve randomly samples 36 bulbs and recorded their life times. The sample mean of the sample is 16 000 hours and the sample standard deviation is 2.8 thousand hours. At the 10% significance level, is there strong enough evidence to support the supplier’s claim?

Compare

test statistic is 2.14
C2 is 1.306

A

Since to is 2.14, it is in the rejection region, therefore we reject Ho

70
Q

The supplier claims that the main life time of their light bulbs is more than 15 000 hours. Steve randomly samples 36 bulbs and recorded their life times. The sample mean of the sample is 16 000 hours and the sample standard deviation is 2.8 thousand hours. At the 10% significance level, is there strong enough evidence to support the supplier’s claim?

Interpret

Since to is 2.14, it is in the rejection region, therefore we reject Ho

A

At the 10% significance level, the data provides sufficient evidence to support the supplier’s claim

71
Q

What are the 6 steps to a hypothesis test?

A
  1. Set up hypotheses
  2. Check assumptions
  3. Determine significance level and find critical value(s)
  4. Calculate the test statistic
  5. Compare the test statistic and critical value(s)
  6. Interpret the results in context of the question

STATS 151 (13 decks)

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