Question

malina and madeline love pepperoni pizza, but sometimes they are disappointed with the small number of pepperonis on their pizza. to investigate, they went to their favorite pizza restaurant at 10 randomly selected times during the week and ordered a large pepperoni pizza. here are the number of pepperonis on each pizza: 47, 36, 25, 37, 46, 36, 49, 32, 32, 34. construct and interpret a 95% confidence interval for the true mean number of pepperonis on a large pizza at this restaurant. state: 95% ci for μ = the true mean number of pepperonis on a large pizza at this restaurant. plan: one - sample t interval for μ. random? random sample of 10 pizzas. normal/large sample? the sample size is small, but the dotplot doesnt show any outliers or strong skewness. the conditions for inference are met. do: (round to 3 decimal places.)

Explanation:

Step1: Calculate sample mean

Find average of the data set.
$\bar{x} = \frac{47+36+25+37+46+36+49+32+32+34}{10} = \frac{374}{10} = 37.4$

Step2: Calculate sample standard deviation

Compute squared deviations, average, then root.
First, squared deviations from mean:
$(47-37.4)^2=92.16$, $(36-37.4)^2=1.96$, $(25-37.4)^2=153.76$, $(37-37.4)^2=0.16$, $(46-37.4)^2=73.96$, $(36-37.4)^2=1.96$, $(49-37.4)^2=134.56$, $(32-37.4)^2=29.16$, $(32-37.4)^2=29.16$, $(34-37.4)^2=11.56$
Sum of squared deviations: $92.16+1.96+153.76+0.16+73.96+1.96+134.56+29.16+29.16+11.56=528.4$
Sample variance: $s^2 = \frac{528.4}{10-1} \approx 58.711$
Sample standard deviation: $s = \sqrt{58.711} \approx 7.662$

Step3: Find t-critical value

For 95% CI, $df=10-1=9$, $t^*=2.262$

Step4: Calculate margin of error

Compute ME using t*, s, n.
$ME = t^* \times \frac{s}{\sqrt{n}} = 2.262 \times \frac{7.662}{\sqrt{10}} \approx 2.262 \times 2.423 \approx 5.481$

Step5: Compute confidence interval

Subtract and add ME to sample mean.
Lower bound: $37.4 - 5.481 = 31.919$
Upper bound: $37.4 + 5.481 = 42.881$

Answer:

31.919 to 42.881

Interpretation (for context):

We are 95% confident that the true mean number of pepperonis on a large pizza at this restaurant is between 31.919 and 42.881.