LRA 8-2

Load the necessary packages: $~$

library(readr)
library(pander)
library(ggplot2)
library(BSDA)

$~$

Problem 1: DDonuts claims that the waiting time of customers for service is normally distributed with a mean of three minutes and a standard deviation of one minute. The quality assurance department found in a sample of $50$ customers that the mean waiting time is 2.85 minutes. At a $0.05$ level of significance, can we conclude that the mean waiting time is less than three minutes?

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Problem: Is the mean waiting time of customers for service at DDonuts less than $3$ minutes?
Ho: The mean waiting time of customers for service at DDonuts is greater than or equal to $3$ minutes.
Ha: The mean waiting time of customers for service at DDonuts is less than $3$ minutes.
Significance level: $0.05$.
Test-statistic: $Z$-statistic since popn standard deviation is known.
Rejection Region:

## [1] -1.644854

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Decision Rule: Reject Ho if z-computed is less than or equal to $-1.645$. Otherwise, fail to reject Ho.

Computation:

pander(zsum.test(mean.x = 2.85, sigma.x = 1, n.x = 50, alternative = "less", mu = 3))

One-sample z-Test: `Summarized x`
Test statistic	P value	Alternative hypothesis	mean of x
-1.061	0.1444	less	2.85

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Decision: Fail to reject Ho since Z-computed is greater than $-1.645$.
Conclusion: There is sufficient data supporting Ho. Hence, the mean waiting time of customers for service at DDonuts is greater than or equal to $3$ minutes.

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Problem 2: A business travel magazine wants to classify transatlantic gateway airports according to the mean rating for the population of business travelers. A rating scale with a low score of $0$ and a high score of $10$ will be used, and airports with a population mean rating greater than $7$ will be designated as superior service airports. The magazine staff surveyed a sample of $60$ business travelers at each airport to obtain the ratings data. The sample for London’s Heathrow Airport provided a sample mean rating of $7.25$ and a sample standard deviation of $1.052$. Do the data indicate that Heathrow should be deisgnated as a superior service airport?

$~$

Problem: Is the mean rating of Heathrow Airport greater than $7$?
Ho: The mean rating of Heathrow Airport is less than or equal to $7$.
Ha: The mean rating of Heathrow Airport is greater than $7$.
Significance level: $0.05$.
Test-statistic: $t$-statistic since the popn variance or standard deviation is unknown.
Critical Region:

## [1] 1.671093

$~$ Decision Rule: reject Ho if t-computed is greater than or equal to $1.671$. Otherwise, fail to reject Ho.

Computation:

pander(tsum.test(mean.x = 7.25, s.x = 1.052, n.x = 60, alternative = "greater", mu = 7))

One-sample t-Test: `Summarized x` $~$
Test statistic	df	P value	Alternative hypothesis	mean of x
1.841	59	0.03534 *	greater	7.25

Decision: Reject Ho since t-computed is greater than $1.671$.
Conclusion: There is no sufficient data supporting Ho. Hence, the mean rating of Heathrow Airport is greater than $7$. This would then further indicate that there is enough data to indicate that Heathrow Airport should be designated as a superior service airport.

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Problem 3: A consumer group complains that the gas tax (in Php per gallon) levied by the local government is too high. The data on gas tax obtained in $30$ urban areas in the country is contained in the “tax.csv” file. Assuming normality, test at the $5\%$ level of significance that the mean gas tax is no greater than $50$ (Php per gallon).

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Problem: Is the mean gas tax no greater (lesser than or equal to) $Php50$ per gallon?
Ho: The mean gas tax is less than or equal to $Php50$ per gallon.
Ha: The mean gas tax is greater than $Php50$ per gallon.
Significance level: $0.05$.
Test statistic: $t$-statistic (since popn variance or standard deviation is unknown)
Computation using R:

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# Import "tax.csv" file:
taxdata <- read.csv("tax.csv")
head(taxdata)

##   Urban.Area tax
## 1          1  53
## 2          2  48
## 3          3  42
## 4          4  57
## 5          5  51
## 6          6  50

#Perform the t-test:
pander(t.test(taxdata$tax, mu=50, alternative="greater"))

One Sample t-test: `taxdata$tax`
Test statistic	df	P value	Alternative hypothesis	mean of x
-1.695	29	0.9496	greater	48.4

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Decision: Fail to reject Ho since $p > 0.05$.
Conclusion: There is sufficient data supporting Ho. Hence, the mean gas tax is not significantly greater than $Php50$ per gallon.

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Problem 4: The managing officer of FlantAsia, an ornamental plant distributor in East Asia claim that their company’s sales decreased by $20\%$ during the month of December. The officer summons the company’s analyst to investigate the claim. The analyst gathers a record of the product sales during the month of December for the past $40$ years. The distribution of the percentage decrease in sales is in the file “decrease.csv”. At the $5\%$ significance level, what would you say about the officer’s claim?

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Problem: Is the mean amount of decrease of FlanAsia sales $20\%$?
Ho: The mean amount of decrease of FlantAsia sales is $20\%$.
Ha: The mean amount of decrease of FlantAsia sales is not $20\%$.
Significance Level: $0.05$.
Test-Statistic: $t$-statistic since the popn variance or standard deviation is unknown.
Computation using R:

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# Import "decrease.csv" file:
decreasedata <- read.csv("decrease.csv")
head(decreasedata)

##   Year decrease
## 1 1981    15.72
## 2 1982    21.24
## 3 1983    23.25
## 4 1984    19.80
## 5 1985    20.90
## 6 1986    17.34

# Perform the t-test:
pander(t.test(decreasedata$decrease, mu=20, alternative="two.sided"))

One Sample t-test: `decreasedata$decrease` $~$
Test statistic	df	P value	Alternative hypothesis	mean of x
-0.7429	39	0.462	two.sided	19.32

Decision: Fail to reject Ho since $p > 0.05$.
Conclusion: There is sufficient data supporting Ho. Hence, the mean amount of decrease of FlantAsia sales is $20\%$.

LRA 8-2

AE311