Question

Boys of a certain age in the nation have an average weight of 85 with a variance of 114.49 lb. A complaint is made that boys are overfed in a municipal children's home. As evidence, a sample of 21 boys of the given age is taken from the children's home with an average weight of 92.5 lb. What can be concluded with α = 0.05?

a) What is the appropriate test statistic?
---Select--- na z-test One-Sample t-test Independent-Samples t-test Related-Samples t-test

b)
Population:
---Select--- feeding method children's home weight boys from the home boys in the nation
Sample:
---Select--- feeding method children's home weight boys from the home boys in the nation

c) Compute the appropriate test statistic(s) to make a decision about H₀.
(Hint: Make sure to write down the null and alternative hypotheses to help solve the problem.)
critical value =  ; test statistic =  
Decision:  ---Select--- Reject H0 Fail to reject H0

d) If appropriate, compute the CI. If not appropriate, input "na" for both spaces below.
[  ,  ]

e) Compute the corresponding effect size(s) and indicate magnitude(s).
If not appropriate, input and select "na" below.
d =  ;   ---Select--- na trivial effect small effect medium effect large effect
r² =  ;   ---Select--- na trivial effect small effect medium effect large effect

f) Make an interpretation based on the results.

The average weight of boys in the municipal children's home was significantly higher than the average weight of boys in the nation.The average weight of boys in the municipal children's home was significantly lower than the average weight of boys in the nation.    The average weight of boys in the municipal children's home was not significantly different than the average weight of boys in the nation.

Answer 1

a) What is the appropriate test statistic?

Answer: z-test

Here, we have to use one sample z test for the population mean because we are given the value for the population standard deviation or population variance. If we are not given the population standard deviation or variance, then we use the t test instead of z test.

b)
Population: boys in the nation

Sample: boys from the children's home

c) Compute the appropriate test statistic(s) to make a decision about H₀.

Here, we have to use one sample z test for the population mean. The null and alternative hypotheses for this test are given as below:

Null hypothesis: H₀: Boys are not overfed in municipal children’s home.

Alternative hypothesis: H_a: Boys are overfed in municipal children’s home.

H₀: µ = 85 versus H_a: µ>85

This is an upper tailed or right tailed (one tailed) test.

The level of significance is given as α = 0.05.

The test statistic formula is given as below:

Z = (Xbar - µ)/[σ/sqrt(n)]

We are given

Xbar = 92.5

µ = 85

σ^2 = 114.49

σ = sqrt(114.49)

σ = 10.7

n = 21

Z = (92.5 – 85)/[10.7/sqrt(21)]

Z = 3.2121

Test statistic = 3.2121

Critical value = 1.6449

(by using z-table)

P-value = 0.0007

(by using z-table)

Test statistic Z > Critical value

P-value < α = 0.05

So, we reject the null hypothesis

Decision: Reject H₀

There is sufficient evidence to conclude that Boys are overfed in municipal children’s home.

d) If appropriate, compute the CI. If not appropriate, input "na" for both spaces below.

Confidence interval for Population mean is given as below:

Confidence interval = Xbar ± Z*σ/sqrt(n)

Xbar = 92.5

σ = 10.7

n = 21

Confidence level = 95%

Z = 1.96

(by using z-table)

Confidence interval = Xbar ± Z*σ/sqrt(n)

Confidence interval = 92.5 ± 1.96*10.7/sqrt(21)

Confidence interval = 92.5 ± 1.96*2.3349

Confidence interval = 92.5 ± 4.5764

Lower limit = 92.5 - 4.5764 = 87.9236

Upper limit = 92.5 + 4.5764 = 97.0764

Confidence limit = (87.9236, 97.0764)

Part e

Formula for effect size is given as below:

d = (Xbar - µ)/σ

d = (92.5 - 85)/10.7

d = 0.700935

medium effect

Part f

We reject the null hypothesis

Decision: Reject H₀

There is sufficient evidence to conclude that Boys are overfed in municipal children’s home.

The average weight of boys in the municipal children's home was significantly higher than the average weight of boys in the nation.