Chapter 4 Foundations for 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.

  1. What is the point estimate for the average height of active individuals? What about the median? average = 171.1 median = 170.3

  2. What is the point estimate for the standard deviation of the heights of active individuals? What about the IQR? SD = 9.4 IQR = 14

  3. 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. 180cm tall is not unusal since it is close to the mean 155cm tall is unusually short since it is close to the min

  4. 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. It is highly unlikely that 2 random samples will be identical in mean and SD. I expect each random sample will be receive different observations due to variation in the population.

  5. 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? Compute this quantity using the data from the original sample under the condition that the data are a simple random sample.

# The measure used to quantify the variability is standard error (SE).

SE <- 9.4/sqrt(507)
SE
## [1] 0.4174687
4.14 Thanksgiving spending, Part I. The 2009 holiday retail season, which kicked o↵ 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% confidence 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% confident that the average spending of these 436 American adults is between $80.31 and $89.11. False - The confidence interval refers to how certain we are that the true population mean is within 80.31 and 89.11

  2. This confidence interval is not valid since the distribution of spending in the sample is right skewed. False - the sample size is large enough to counteract the skew

  3. 95% of random samples have a sample mean between $80.31 and $89.11. False - The confidence interval refers to how certain we are that the true population mean is within 80.31 and 89.11

  4. We are 95% confident that the average spending of all American adults is between $80.31 and $89.11. True

  5. A 90% confidence interval would be narrower than the 95% confidence 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% confidence interval to a third of what it is now, we would need to use a sample 3 times larger. False - We would need a sample larger than 3 times

  7. The margin of error is 4.4. True

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 identified 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 first counted to 10 successfully. Also provided are some sample statistics.
  1. Are conditions for inference satisfied? Yes - sample is random. sample size > 30 and the distribution is not skewed.
  1. Suppose you read online that children first 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 fist count to 10 successfully is less than the general average of 32 months. Use a significance level of 0.10.
se <- 4.31/sqrt(36)
z = (30.69 -32)/se
p = pnorm(z)
p
## [1] 0.0341013

  1. Interpret the p-value in context of the hypothesis test and the data. Since the P value is so low, the NULL hypothesis is rejected

  2. Calculate a 90% confidence interval for the average age at which gifted children first count to 10 successfully.

low <- 30.69 - 1.65 * se
high <- 30.69 + 1.65 * se
low
## [1] 29.50475
high
## [1] 31.87525
  1. Do your results from the hypothesis test and the confidence interval agree? Explain. Yes – we are 90% certain gifted children first count to 10 between 29.5 and 31.9 months old

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.

(a)Perform a hypothesis test to evaluate if these data provide convincing evidence that the average IQ of mothers of gifted children is different than the average IQ for the population at large, which is 100. Use a significance level of 0.10.

se <- 6.5/sqrt(36)
z <- (118.2 - 100)/se
p <- (pnorm(z, mean = 0, sd = 1, lower.tail = FALSE) * 2)
p
## [1] 2.44044e-63

(b)Calculate a 90% confidence interval for the average IQ of mothers of gifted children.

low <- 118.2 - 1.65 * se
high <- 118.2 + 1.65 * se
low
## [1] 116.4125
high
## [1] 119.9875

(c)Do your results from the hypothesis test and the confidence interval agree? Explain. Yes – we are 90% certain the average IQ of mothers of gifted children is between 116.4125 and 119.9875

4.34 CLT. Define 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 describes the shape of the means of multiple samples from a population, of the same size. As the sample size increases, the shape becomes more normal, the center moves closer to the population mean, and the spread becomes narrower.

4.40 CFLBs. A manufacturer of compact fluorescent 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.

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

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

(b)Describe the distribution of the mean lifespan of 15 light bulbs. Since the sample is so small the spread is wide and is not very normal

bulb15 <-  rnorm(15, mean = 9000, sd = 1000)
hist(bulb15)

(c)What is the probability that the mean lifespan of 15 randomly chosen light bulbs is more than 10,500 hours? Zero

Z <- (10500 - 9000)/258
p <- 1 - pnorm(Z)
p
## [1] 3.050719e-09

(d)Sketch the two distributions (population and sampling) on the same scale.

hist(rnorm(9000, 1000), col=rgb(1,0,1,0.5))
hist(bulb15,  col=rgb(1,1,1,0.5), add=T)
box()

(e)Could you estimate the probabilities from parts (a) and (c) if the lifespans of light bulbs had a skewed distribution? No – We need a sample more than 30 and a normal distribution.

4.48 Same observation, di↵erent 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.

__ Sample size increases = P-Value decreases. The sample size increases the Z-score which causes the p-value to decrease.__