Chapter 4 - Foundations of Inference
Practice: 4.3, 4.13, 4.23, 4.25, 4.39, 4.47
Graded: 4.4, 4.14, 4.24, 4.26, 4.34, 4.40, 4.48

4.4 Heights of adults

Researchers studying anthropometry collected body girth measurements and skeletal diameter measurements, as well as age, weight, height and gender, for 507 physically active individuals. The histogram below shows the sample distribution of heights in centimeters.

(a) What is the point estimate for the average height of active individuals? What about the median?

Point estimate = 171.1 Median = 170.3
(b) What is the point estimate for the standard deviation of the heights of active individuals? What about the IQR?

SD = 9.4 IQR =

177.8-163.8
## [1] 14


(c) Is a person who is 1m 80cm (180 cm) tall considered unusually tall? And is a person who is 1m 55cm (155cm) considered unusually short? Explain your reasoning.

We will see how many standard deviations from the mean these heights are.

sd4.4 = 9.4
mean4.4 = 171.1

#180 check
(180 - mean4.4)/sd4.4
## [1] 0.9468085
#155 check
(155 - mean4.4)/sd4.4
## [1] -1.712766

Both observations are within two standard deviations of the mean, so neither are unusual by that definition. The 155 cm person is more short than the 180 cm person is tall.
(d) The researchers take another random sample of physically active individuals. Would you expect the mean and the standard deviation of this new sample to be the ones given above? Explain your reasoning.

We would expect the mean and standard deviations to be similar though not exact, assuming a sample size of at least 30.
(e) The sample means obtained are point estimates for the mean height of all active individuals, if the sample of individuals is equivalent to a simple random sample. What measure do we use to quantify the variability of such an estimate (Hint: recall that SD¯ x = pn)? Compute this quantity using the data from the original sample under the condition that the data are a simple random sample. 

sd4.4/(sqrt(507))
## [1] 0.4174687



4.14 Thanksgiving spending, Part I

The 2009 holiday retail season, which kicked off on November 27, 2009 (the day after Thanksgiving), had been marked by somewhat lower self-reported consumer spending than was seen during the comparable period in 2008. To get an estimate of consumer spending, 436 randomly sampled American adults were surveyed. Daily consumer spending for the six-day period after Thanksgiving, spanning the Black Friday weekend and Cyber Monday, averaged $84.71. A 95% con???dence interval based on this sample is ($80.31, $89.11). Determine whether the following statements are true or false, and explain your reasoning.

  1. We are 95% con???dent that the average spending of these 436 American adults is between $80.31 and $89.11.

    False. We’re 100% sure that point estimate is in the confidence interval.
  2. This con???dence interval is not valid since the distribution of spending in the sample is right skewed.

    False. Because the sample size is greater than 30, skew doesn’t matter, and the sample mean distribution will be nearly normal.
  3. 95% of random samples have a sample mean between $80.31 and $89.11.

    False. Confidence intervals do not relate to a sample mean.
  4. We are 95% con???dent that the average spending of all American adults is between $80.31 and $89.11.

    True.
  5. A 90% con???dence interval would be narrower than the 95% con???dence interval since we don’t need to be as sure about our estimate.
    True.
  6. In order to decrease the margin of error of a 95% con???dence interval to a third of what it is now, we would need to use a sample 3 times larger.

    False. It would need to be 9 (3^2) ties larger.
  7. The margin of error is 4.4.
    TRUE.
    Margin of error:
(89.11-80.31)/2
## [1] 4.4


4.24 Gifted children, Part I

Researchers investigating characteristics of gifted children collected data from schools in a large city on a random sample of thirty-six children who were identi???ed as gifted children soon after they reached the age of four. The following histogram shows the distribution of the ages (in months) at which these children ???rst counted to 10 successfully. Also provided are some sample statistics.

  1. Are conditions for inference satis???ed?
    Probably. Sample size is greater than 30. It could go either way, but the sample of 36 is likely fewer than 10% of the total population. Selection was random - at least random as far as picking from kids who were identified in some way as gifted, which could be a process subject to bias. Sample looks to be relatively normallly distributed.
  2. Suppose you read online that children ???rst count to 10 successfully when they are 32 months old, on average. Perform a hypothesis test to evaluate if these data provide convincing evidence that the average age at which gifted children ???st count to 10 successfully is less than the general average of 32 months. Use a signi???cance level of 0.10.

    h0 = 32 ha < 32

Get the z score.

mean_4.24 <- 30.69
n_4.24 <- 36
sd_4.24 <- 4.31
x_4.24 <- 32


SE_4.24 <- sd_4.24/sqrt(n_4.24)
Z_4.24 <- (mean_4.24 - x_4.24)/(SE_4.24)
#one-side test
p_4.24 <- pnorm(Z_4.24, mean = 0, sd = 1) 

p_4.24
## [1] 0.0341013


(c) Interpret the p-value in context of the hypothesis test and the data.

Since our p-value is less than the significance level of .1, we fail to reject the null hypothesis that gifted children learn to count to 10 at the same age as all children.
(d) Calculate a 90% con???dence interval for the average age at which gifted children ???rst count to 10 successfully.

#1.645 is z critical value for .90 CI
#lower
mean_4.24 - 1.645 * SE_4.24
## [1] 29.50834
#upper
mean_4.24 + 1.645 * SE_4.24
## [1] 31.87166


(e) Do your results from the hypothesis test and the con???dence interval agree? Explain.
They do. The confidence interval says that there’s a 90% chance the true mean for gifted children is between 29.5 and 31.9 months. 32 months is outside that range.

4.26 Gifted children, Part II.

Exercise 4.24 describes a study on gifted children. In this study, along with variables on the children, the researchers also collected data on the mother’s and father’s IQ of the 36 randomly sampled gifted children. The histogram below shows the distribution of mother’s IQ. Also provided are some sample statistics.

  1. Perform a hypothesis test to evaluate if these data provide convincing evidence that the average IQ of mothers of gifted children is di???erent than the average IQ for the population at large, which is 100. Use a signi???cance level of 0.10.

    h0 = 100 ha <> 100

Get the z score.

mean_4.26 <- 118.2
n_4.26 <- 36
sd_4.26 <- 6.5
x_4.26 <- 100


SE_4.26 <- sd_4.26/sqrt(n_4.26)
Z_4.26 <- (mean_4.26 - x_4.26)/(SE_4.26)
#two-sided
p_4.26 <- pnorm(Z_4.26, mean = 0, sd = 1, lower.tail = FALSE) * 2 

p_4.26
## [1] 2.44044e-63

This is a tiny p-value, much lower than the significance level of 0.10. We reject h0.
(b) Calculate a 90% con???dence interval for the average IQ of mothers of gifted children.

#1.645 is z critical value for .90 CI
#lower
mean_4.26 - 1.645 * SE_4.26
## [1] 116.4179
#upper
mean_4.26 + 1.645 * SE_4.26
## [1] 119.9821


(c) Do your results from the hypothesis test and the con???dence interval agree? Explain.
Heck ya, they do. The lower tail of the confidence interval is considerably greater than the average IQ of the overall population.

4.34 CLT.

De???ne the term “sampling distribution” of the mean, and describe how the shape, center, and spread of the sampling distribution of the mean change as sample size increases.

Sampling distribution of the mean conveys how means from multiple samples of a population are distributed. As sample sizes increase, the sampling distribution of the mean will represent a normal distribution.

4.40 CFLBs

A manufacturer of compact ???uorescent light bulbs advertises that the distribution of the lifespans of these light bulbs is nearly normal with a mean of 9,000 hours and a standard deviation of 1,000 hours.

  1. What is the probability that a randomly chosen light bulb lasts more than 10,500 hours?

p_4.40 <- (1 - pnorm(10500, mean = 9000, sd = 1000))
p_4.40
## [1] 0.0668072

6.69%
(b) Describe the distribution of the mean lifespan of 15 light bulbs.

The distribution of the lifepsans of the light bulbs population is nearly normal, so the mean lifespan distribution should become more normal as it grows. Still, 15 light bulbs is smaller than 30 and could be affected by a few outliers.
(c) What is the probability that the mean lifespan of 15 randomly chosen light bulbs is more than 10,500 hours?

p_4.40c <- round(1-pnorm((10500-9000)/(1000/sqrt(258))),6)
p_4.40c
## [1] 0

Odds are virtually 0.
(d) Sketch the two distributions (population and sampling) on the same scale. 

library(DATA606)
## Loading required package: shiny
## Loading required package: openintro
## Please visit openintro.org for free statistics materials
## 
## Attaching package: 'openintro'
## The following objects are masked from 'package:lattice':
## 
##     ethanol, lsegments
## The following objects are masked from 'package:datasets':
## 
##     cars, chickwts, trees
## Loading required package: OIdata
## Loading required package: RCurl
## Loading required package: bitops
## Loading required package: maps
## Warning: package 'maps' was built under R version 3.3.3
## Loading required package: ggplot2
## 
## Attaching package: 'ggplot2'
## The following object is masked from 'package:openintro':
## 
##     diamonds
## Loading required package: markdown
## Warning: package 'markdown' was built under R version 3.3.3
## 
## Welcome to CUNY DATA606 Statistics and Probability for Data Analytics 
## This package is designed to support this course. The text book used 
## is OpenIntro Statistics, 3rd Edition. You can read this by typing 
## vignette('os3') or visit www.OpenIntro.org. 
##  
## The getLabs() function will return a list of the labs available. 
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## The demo(package='DATA606') will list the demos that are available.
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## Attaching package: 'DATA606'
## The following object is masked from 'package:utils':
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##     demo
#Population
normalPlot(9000,1000)


#Sampling
normalPlot(9000,(1000/sqrt(15)))


(e) Could you estimate the probabilities from parts (a) and (c) if the lifespans of light bulbs had a skewed distribution?
Nope, you need a normal distribution.

4.48 Same observation, different sample size.


Suppose you conduct a hypothesis test based on a sample where the sample size is n = 50, and arrive at a p-value of 0.08. You then refer back to your notes and discover that you made a careless mistake, the sample size should have been n = 500. Will your p-value increase, decrease, or stay the same? Explain.
P value will decrease, as a larger sample means a smaller standard error, decreasing the Z value.