Textbook Prices

Will you save money buying textbooks from Amazon instead of the UCLA bookstore?

• Data: 68 textbooks with prices at both locations
• This is PAIRED: Same book measured at two stores
• Variable: price_diff = UCLA price − Amazon price

Research question: On average, do UCLA bookstore prices differ from Amazon prices?

ucla_textbooks_f18

# A tibble: 68 × 5
   subject                 course_num bookstore_new amazon_new price_diff
   <fct>                   <fct>              <dbl>      <dbl>      <dbl>
 1 American Indian Studies M10                48.0       47.4       0.520
 2 Anthropology            2                  14.3       13.6       0.710
 3 Arts and Architecture   10                 13.5       12.5       0.97 
 4 Asian                   M60W               49.3       55.0      -5.69 
 5 Astronomy               4                 120.       125.       -4.83 
 6 Communication           10                 17.0       11.8       5.18 
 7 Comparative Literature  2CW                12.0       10.9       1.09 
 8 Dance                   10                 26.8       38.9     -12.2  
 9 English                 19                  9.96       8.99      0.97 
10 English Composition     1A                 40.0       35         4.97 
# ℹ 58 more rows

Side-by-side Boxplots

Here are side-by-side dotplots

Paired data

• One way to analyze the data would be to treat the books on Amazon and the books at the bookstore as two groups. Then we could compare the difference in the group means as we did in Chapter 20
• Each observation would be a book on Amazon or a book at the bookstore
• However, this ignores the paired structure of the data (observations are not independent)
• Such analysis would not use all available information and will have lower power

EDA: Textbook Price Differences

On average, UCLA bookstore prices are $3.58 higher than Amazon.

Hypotheses for Textbook Prices

Let \(\mu_d\) = mean price difference (UCLA − Amazon) for all textbooks

Hypotheses:

• \(H_0: \mu_d = 0\) (no difference in prices on average)
• \(H_A: \mu_d \neq 0\) (prices differ on average)

This is a two-sided test.

Checking Conditions

Independence:

• Random sample of textbooks ✓
• Each textbook is independent of others ✓

Normality:

• n = 68 ≥ 30 ✓
• The data are skewed and there are outliers

We will proceed cautiously with using the t-distribution, but randomization would be preferred

Test Statistic

Degrees of freedom: \(df = n - 1 = 68 - 1 = 67\)

Standard error:

\[SE = \frac{s_d}{\sqrt{n}} = \frac{13.42}{\sqrt{68}} = 1.63\]

T-statistic:

\[T = \frac{\bar{x}_d - 0}{SE} = \frac{3.58 - 0}{1.63} = 2.2\]

Calculating the P-value

Use a t-distribution (df = 67).

P-value = 0.0312 (from Jamovi)

Conclusion: Hypothesis Test

Results:

• T = 2.2
• P-value = 0.0312
• Using α = 0.05: P-value < 0.05

Decision: Reject \(H_0\)

Conclusion: The data provide convincing evidence that, on average, UCLA bookstore prices differ from Amazon prices for textbooks. However, we should be cautious given the skew and outliers present in the data and should verify our results using randomization.

CI for Mean Difference

Given: \(\bar{x}_d = 3.58\), \(SE = 1.63\), \(df = 67\)

Critical value: \(t^*_{67} = 1.996\) (from Jamovi)

95% CI:

\[3.58 \pm 1.996 \times 1.63 = (0.33, 6.83)\]

Interpreting the CI

95% CI: ($0.33, $6.83)

Interpretation: We are 95% confident that the mean price difference (UCLA − Amazon) for all textbooks is between $0.33 and $6.83.

Does CI include 0? No

This is consistent with our hypothesis test—we rejected \(H_0\), and 0 is not in the CI.

Temperature

• Data set temperature collected temperature readings from 32 NASA-GISS stations based on a random sample of latitude-longitude coordinates.
• Variable Past represents average reading from 1901 through 1950 and variable Recent is the recorded average temprature from 1951 through 2000
• This a matched-pairs design since we are comparing temperatures from the same locations at different time periods, and the differences can be analyzed to determine if we have statistically significant evidence of the rise of the temperature.