Problem 4.4 - Height of Adults

  1. Average height of active individuals is 171.9 and median height is 170.3
  2. SD is 9.4 and IQR is 14 (177.8 - 163.8)
z180 <- (180 - 171.1) / 9.4
z180
## [1] 0.9468085
z155 <- (155 - 171.1) / 9.4
z155
## [1] -1.712766

As per the answers above, it seems that both z180 and z155 are 0.94 and -1.71 which are within one standard deviation which does not show that they are unusual. There is no standard that what is starting point for tall or vice versa that’s why as per the results we can’t tell that they are either tall or short but what we can tell is that they fall within 2 standard deviations and is not unusual.

  1. I would not expect same results due to change in the new sample.
  2. Measure of quantifying the variability is:
sd <- 9.4
n <- sqrt(507)
sd/n
## [1] 0.4174687

Standard error of mean is 0.417.

Problem 4.14 - Thanksgiving spending

  1. False - Inference is always measured on the population parameter. Means that if we take different samples and measure their means, we will be 95 percent confident that the their means will be near population mean.
  2. False - Sample size is more than 30 and with the sample size of 436 observations, the sample size is sufficient enough to justify the distribution.
  3. False - Confidence intervals are not about random sample means
  4. True - The statement is correct as confidence level is for average mean of population
  5. True - As the confidence level is decreased, we would be less sure if the mean is close to the population mean.
  6. False - If we decrease the marginal error by third we have to increase sample by 3^2 = 9 times
  7. True
SE <- (89.11 - 84.71) /1.96
1.96 * SE
## [1] 4.4

Problem 4.24 - Gifted Children

  1. Yes, sample size is more than 30 and the sample of 36 from large city is obviously under 10 percent of entire population. Data does not seem strongly skewed. All conditions are satisfied for inference
  2. Ho = 32 Ha < 32
SE2 <- 4.31/sqrt(36)
SE2
## [1] 0.7183333
z2 <- (30.69 - 32) / 0.7183333
z2
## [1] -1.823666
p = pnorm(z2)
p
## [1] 0.03410129

P value is 0.0341 which is less than 0.10 hence we reject Ho in the favor of Ha.

  1. As the p-value is low that is an evidence against rejecting Ho and vice versa.
x1 <- 30.69 - 1.65 * SE2
x2 <- 30.60 + 1.65 * SE2
x1
## [1] 29.50475
x2
## [1] 31.78525

The confidence interval for 90 percent is 29.50 and 31.78

  1. Yes, our results agree with the hypothesis results. We are 90 percent confident that average mean should be within 29.50 and 31.78 which is less than 32 months (Ha < 32)

Problem 4.26 - Gifted Children Part 2

  1. Ho = 100 Ha is not equal to 100
SE3 <- 6.5/sqrt(36)
SE3
## [1] 1.083333
zIQ <- (118.2 - 100) / 1.08333
zIQ
## [1] 16.80005

With the z-score of 16.8, the p-value would be close to 0 and we will reject the null hypothesis.

118.2 + 1.65 * SE3
## [1] 119.9875
118.2 - 1.65 * SE3
## [1] 116.4125

The confidence interval for 90% is 116.4125 and 119.9875

  1. Yes, results say that the average IQ of mothers should be between 116.41 and 119.98 with 90 percent confidence interval which supports the hypothesis Ha stating that the population average is not equal to 100.

Problem 4.34 - CLT

The sampling distribution of mean is actually the distribution of the sample means taken from different samples. Central Limit Theorem states that the distribution of sample means is approximately normal. Approximation gets better with the increasein sample size.

Problem 4.40 - CFLBs

x3 <- (10500 - 9000) / 1000
1-pnorm(1.5)
## [1] 0.0668072

Probability of bulb lasts more than 10,500 hours is 0.0668072.

N(mu, SD/sqrt(n)) N(9000,258.1989)

(10500 - 9000) / 258.1989
## [1] 5.809475
1-pnorm(5.81)
## [1] 3.123642e-09

Probability is less than 0 which makes it almost 0.

  1. Black is population distribution while green is sampling distribution.
s <- seq(5000,13000,0.01)
plot(s, dnorm(s,9000, 1000), type="l", ylim = c(0,0.002), ylab = "", xlab = "Lifespan (hours)")
lines(s, dnorm(s,9000, 258.1989), col="green")

  1. No we cannot estimate without normal distribution in part (a) and especially part (c) with such small sample size.

Problem 4.48 - Same observation, different sample size

P-value will be decreased if sample size gets larger and vice versa