Sampling Distribution of z

The distribution of z values that would be obtained if a value of z were calculated for each sample mean for all possible random samples of a given size from some population.

How to convert a raw score into a standard score (z) ?

standard score = (raw score - mean) / standard deviation

How to convert a sample mean to z?

z = [ ̅X - µ(hyp)] / σ(sub ̅X )

z Test for a Population Mean

A hypothesis test that evaluates how far the observed sample mean deviates, in standard error units, from the hypothesized population.

Calculate the value of the z test for the following situation:

̅X = 566

σ = 30

n = 36

µ(hyp) = 560

z = [ ̅X - µ(hyp)] / [σ / √n]

z = (566 - 560) / (30/√5) = 6/5 = 1.2

Calculate the value of the z test for the following situation:

̅X = 24

σ = 4

n = 64

µ(hyp) = 25

z = [ ̅X - µ(hyp)] / [σ / √n]

z = (24 - 25) / (4/√64) = -1 / 0.5 = -2

Calculate the value of the z test for the following situation:

̅X = 82

σ = 14

n = 49

µ(hyp) = 75

z = [ ̅X - µ(hyp)] / [σ / √n]

z = (82 - 75) / (14/√49) = 7 / 2 = 3.5

Calculate the value of the z test for the following situation:

̅X = 136

σ = 15

n = 25

µ(hyp) = 146

z = [ ̅X - µ(hyp)] / [σ / √n]

z = (136 - 146) / (15/√25) = -10 / 3 = -3,33

Null Hypothesis (H0)

A statistical hypothesis that usually asserts that noting special is happening with respect to some characteristic of the underlying population.

Alternative Hypothesis (H1)

The opposite of the null hypothesis.

Research Hypothesis

Usually identified with the alternative hypothesis, this is the informal hypothesis or hunch that inspires the entire investigation.

Indicate what’s wrong with the following statistical hypothesis:

H0 : µ = 155

H1 : µ ≠ 160

Different numbers appear in H0 and H1

Indicate what’s wrong with the following statistical hypothesis:

H0 : ̅X = 241

H1 : ̅X ≠ 241

Sample means (rather than population means) appear in H0 and H1.

First using words, then symbols, identify the null hypothesis for the following situation.

A school administrator wishes to determine whether sixth-grade boys in her school district differ, on average, from the national norms of 10.2 pushups for sixth-grade boys.

Sixth-grade boys in her school district average 10.2 pushups.

H0 : µ = 10.2

First using words, then symbols, identify the null hypothesis for the following situation.

A consumer group investigates whether, on average, the true weights of packages of ground beef sold by a large supermarket chain differ from the specified 16 ounces.

On average, weights of packages of ground beef sold by a large supermarket chain equal 16 ounces.

H0 : µ = 16

First using words, then symbols, identify the null hypothesis for the following situation.

A marriage counselor wishes to determine whether, during a standard conflict-resolution session, his clients differ, on average, from the 11 verbal interruptions reported for “well-adjusted couples”.

The marriage counselor’s clients average 11 interruptions per session.

H0 : µ = 11

Decision Rule

Specifies precisely when H0 should be rejected (because the observed z qualifies as a rare outcome).

Critical z Score

A z score that separates common from rare outcomes and hence dictates whether H0 should be retained or rejected.

Level of Significance (α)

The degree of rarity required of an observed outcome in order to reject the null hypothesis (H0).

Indicate whether H0 should be retained or rejected and justify your answer by specifying the precise relationship between observed and critical z scores.

Should H0 be retained or rejected, given a hypothesis test with critical z scores of ± 1.96 and…

z = 1.74 ?

Retain H0 at the 0.5 level of significance because z = 1.74 is less positive than 1.96.

Indicate whether H0 should be retained or rejected and justify your answer by specifying the precise relationship between observed and critical z scores.

Should H0 be retained or rejected, given a hypothesis test with critical z scores of ± 1.96 and…

z = 0.13 ?

Retain H0 at the 0.5 level of significance because z = 0.13 is less positive than 1.96.

Indicate whether H0 should be retained or rejected and justify your answer by specifying the precise relationship between observed and critical z scores.

Should H0 be retained or rejected, given a hypothesis test with critical z scores of ± 1.96 and…

z = -2.51 ?

RejectH0 at the 0.5 level of significance because z = -2.51 is more negative than -1.96.

According to the American Psychological Association, members with a doctorate and a full-time teaching appointment earn, on the average, $82,500 per year, with a standard deviation of $6,000. An investigator wishes to determine whether $82,500 is also the mean salary for all female members with a doctorate and a full-time teaching appointment. Salaries are obtained for a random sample of 100 women from this population, and the mean salary equals $80,100.

Someone claims that the observed difference between $80,100 and $82,500 is large enough by itself to support the conclusion that female members earn less than male members. Explain why it is important to conduct a hypothesis test.

The observed difference between $80,100 and $82,500 cannot be interpreted at face value, as it could have happened just by chance. A hypothesis test permits us to evaluate the effect of chance by measuring the observed difference relative to the standard error of the mean.

According to the American Psychological Association, members with a doctorate and a full-time teaching appointment earn, on the average, $82,500 per year, with a standard deviation of $6,000. An investigator wishes to determine whether $82,500 is also the mean salary for all female members with a doctorate and a full-time teaching appointment. Salaries are obtained for a random sample of 100 women from this population, and the mean salary equals $80,100.

The investigator wishes to conduct a hypothesis test for what population?

All female members of the APA with a Ph.D degree and a full-time teaching appointment.

According to the American Psychological Association, members with a doctorate and a full-time teaching appointment earn, on the average, $82,500 per year, with a standard deviation of $6,000. An investigator wishes to determine whether $82,500 is also the mean salary for all female members with a doctorate and a full-time teaching appointment. Salaries are obtained for a random sample of 100 women from this population, and the mean salary equals $80,100.

What is the null hypothesis, H0?

H0 : µ = 82,500

What is the alternative hypothesis, H1?

H1 : µ ≠ 82,500

a) Specify the decision rule, using the 0.5 level of significance.

b) Calculate the value of z.

c) What is your decision about H0?

d) Using words, interpret this decision in terms of the original problem.

(To calculate z, remember to convert the standard deviation to a standard error).

a) Reject H0 at the .05 level of significance if z ≥ 1.96 or z ≤ -1.96

b) z = [80,000 - 82,500] / [6000/√100] = -2400/600 = -4.00

c) Reject H0 at the .05 level of significance because z = -4.00 is more negative than -1.96.

d) The average salary of all female APA members (with a Ph.D and a full-time teaching appointment) is less than $82,500.