Question

Researchers in a 2015 study by Flannery et al. sought to estimate the effectiveness of the seasonal influenza vaccine in preventing acute respiratory infection (ARI). A total of 507 subjects from Texas who showed symptoms of ARI were recruited for this study. The data provided show the result of each subject's test for influenza.

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Suppose the researchers wish to construct a 95% z ‑confidence interval for the proportion of patients with ARI symptoms who would test positive for influenza. Calculate the lower and upper bounds of this confidence interval. Please specify your answer as a decimal, rounded to at least three decimal places.

lower bound:

upper bound:

Patient" "Influenza"

"1" "Positive"

"2" "Negative"

"3" "Positive"

"4" "Positive"

"5" "Negative"

"6" "Negative"

"7" "Positive"

"8" "Negative"

"9" "Negative"

"10" "Negative"

"11" "Positive"

"12" "Negative"

"13" "Positive"

"14" "Positive"

"15" "Positive"

"16" "Negative"

"17" "Negative"

"18" "Positive"

"19" "Positive"

"20" "Positive"

"21" "Positive"

"22" "Positive"

"23" "Positive"

"24" "Negative"

"25" "Positive"

"26" "Negative"

"27" "Positive"

"28" "Positive"

"29" "Negative"

"30" "Positive"

"31" "Positive"

"32" "Negative"

"33" "Positive"

"34" "Negative"

"35" "Negative"

"36" "Negative"

"37" "Negative"

"38" "Positive"

"39" "Positive"

"40" "Positive"

"41" "Negative"

"42" "Negative"

"43" "Negative"

"44" "Negative"

"45" "Negative"

"46" "Negative"

"47" "Negative"

"48" "Negative"

"49" "Positive"

"50" "Positive"

"51" "Positive"

"52" "Positive"

"53" "Positive"

"54" "Negative"

"55" "Negative"

"56" "Positive"

"57" "Negative"

"58" "Positive"

"59" "Positive"

"60" "Negative"

"61" "Negative"

"62" "Positive"

"63" "Negative"

"64" "Negative"

"65" "Negative"

"66" "Positive"

"67" "Positive"

"68" "Positive"

"69" "Positive"

"70" "Positive"

"71" "Positive"

"72" "Negative"

"73" "Negative"

"74" "Positive"

"75" "Negative"

"76" "Negative"

"77" "Negative"

"78" "Negative"

"79" "Negative"

"80" "Negative"

"81" "Positive"

"82" "Negative"

"83" "Negative"

"84" "Positive"

"85" "Negative"

"86" "Positive"

"87" "Positive"

"88" "Negative"

"89" "Positive"

"90" "Positive"

"91" "Negative"

"92" "Positive"

"93" "Negative"

"94" "Positive"

"95" "Negative"

"96" "Positive"

"97" "Negative"

"98" "Negative"

"99" "Negative"

"100" "Negative"

"101" "Negative"

"102" "Positive"

"103" "Positive"

"104" "Negative"

"105" "Negative"

"106" "Negative"

"107" "Positive"

"108" "Negative"

"109" "Negative"

"110" "Negative"

"111" "Negative"

"112" "Positive"

"113" "Negative"

"114" "Negative"

"115" "Negative"

"116" "Positive"

"117" "Positive"

"118" "Positive"

"119" "Negative"

"120" "Negative"

"121" "Negative"

"122" "Negative"

"123" "Negative"

"124" "Negative"

"125" "Negative"

"126" "Positive"

"127" "Negative"

"128" "Positive"

"129" "Negative"

"130" "Positive"

"131" "Negative"

"132" "Negative"

"133" "Negative"

"134" "Negative"

"135" "Negative"

"136" "Positive"

"137" "Negative"

"138" "Positive"

"139" "Negative"

"140" "Negative"

"141" "Positive"

"142" "Positive"

"143" "Negative"

"144" "Positive"

"145" "Positive"

"146" "Positive"

"147" "Negative"

"148" "Negative"

"149" "Negative"

"150" "Negative"

"151" "Positive"

"152" "Positive"

"153" "Negative"

"154" "Negative"

"155" "Negative"

"156" "Positive"

"157" "Positive"

"158" "Positive"

"159" "Positive"

"160" "Negative"

"161" "Positive"

"162" "Negative"

"163" "Positive"

"164" "Negative"

"165" "Negative"

"166" "Positive"

"167" "Negative"

"168" "Negative"

"169" "Positive"

"170" "Negative"

"171" "Positive"

"172" "Positive"

"173" "Negative"

"174" "Negative"

"175" "Negative"

"176" "Positive"

"177" "Negative"

"178" "Negative"

"179" "Positive"

"180" "Positive"

"181" "Negative"

"182" "Negative"

"183" "Negative"

"184" "Negative"

"185" "Negative"

"186" "Negative"

"187" "Negative"

"188" "Negative"

"189" "Negative"

"190" "Positive"

"191" "Positive"

"192" "Negative"

"193" "Positive"

"194" "Negative"

"195" "Negative"

"196" "Negative"

"197" "Positive"

"198" "Positive"

"199" "Positive"

"200" "Positive"

"201" "Negative"

"202" "Positive"

"203" "Negative"

"204" "Negative"

"205" "Negative"

"206" "Negative"

"207" "Positive"

"208" "Negative"

"209" "Negative"

"210" "Negative"

"211" "Positive"

"212" "Negative"

"213" "Positive"

"214" "Negative"

"215" "Negative"

"216" "Negative"

"217" "Negative"

"218" "Positive"

"219" "Positive"

"220" "Negative"

"221" "Positive"

"222" "Positive"

"223" "Negative"

"224" "Positive"

"225" "Negative"

"226" "Negative"

"227" "Negative"

"228" "Negative"

"229" "Positive"

"230" "Negative"

"231" "Negative"

"232" "Negative"

"233" "Positive"

"234" "Positive"

"235" "Positive"

"236" "Negative"

"237" "Positive"

"238" "Negative"

"239" "Negative"

"240" "Negative"

"241" "Positive"

"242" "Negative"

"243" "Negative"

"244" "Positive"

"245" "Negative"

"246" "Negative"

"247" "Negative"

"248" "Negative"

"249" "Negative"

"250" "Positive"

"251" "Positive"

"252" "Positive"

"253" "Negative"

"254" "Negative"

"255" "Positive"

"256" "Negative"

"257" "Negative"

"258" "Negative"

"259" "Negative"

"260" "Positive"

"261" "Negative"

"262" "Negative"

"263" "Negative"

"264" "Positive"

"265" "Positive"

"266" "Negative"

"267" "Negative"

"268" "Negative"

"269" "Negative"

"270" "Negative"

"271" "Positive"

"272" "Positive"

"273" "Negative"

"274" "Negative"

"275" "Negative"

"276" "Negative"

"277" "Positive"

"278" "Negative"

"279" "Negative"

"280" "Negative"

"281" "Positive"

"282" "Negative"

"283" "Negative"

"284" "Negative"

"285" "Positive"

"286" "Positive"

"287" "Positive"

"288" "Negative"

"289" "Negative"

"290" "Negative"

"291" "Negative"

"292" "Positive"

"293" "Negative"

"294" "Negative"

"295" "Positive"

"296" "Negative"

"297" "Negative"

"298" "Positive"

"299" "Negative"

"300" "Positive"

"301" "Negative"

"302" "Positive"

"303" "Positive"

"304" "Positive"

"305" "Negative"

"306" "Negative"

"307" "Negative"

"308" "Positive"

"309" "Negative"

"310" "Negative"

"311" "Negative"

"312" "Negative"

"313" "Negative"

"314" "Positive"

"315" "Negative"

"316" "Negative"

"317" "Negative"

"318" "Negative"

"319" "Positive"

"320" "Negative"

"321" "Positive"

"322" "Negative"

"323" "Positive"

"324" "Negative"

"325" "Positive"

"326" "Negative"

"327" "Positive"

"328" "Negative"

"329" "Negative"

"330" "Negative"

"331" "Positive"

"332" "Positive"

"333" "Negative"

"334" "Negative"

"335" "Negative"

"336" "Positive"

"337" "Negative"

"338" "Negative"

"339" "Negative"

"340" "Positive"

"341" "Negative"

"342" "Negative"

"343" "Positive"

"344" "Negative"

"345" "Positive"

"346" "Negative"

"347" "Negative"

"348" "Positive"

"349" "Positive"

"350" "Negative"

"351" "Positive"

"352" "Negative"

"353" "Negative"

"354" "Positive"

"355" "Positive"

"356" "Positive"

"357" "Negative"

"358" "Negative"

"359" "Negative"

"360" "Negative"

"361" "Positive"

"362" "Negative"

"363" "Positive"

"364" "Positive"

"365" "Positive"

"366" "Positive"

"367" "Negative"

"368" "Positive"

"369" "Negative"

"370" "Negative"

"371" "Negative"

"372" "Negative"

"373" "Positive"

"374" "Negative"

"375" "Negative"

"376" "Negative"

"377" "Negative"

"378" "Negative"

"379" "Positive"

"380" "Positive"

"381" "Negative"

"382" "Positive"

"383" "Negative"

"384" "Positive"

"385" "Positive"

"386" "Negative"

"387" "Negative"

"388" "Positive"

"389" "Negative"

"390" "Negative"

"391" "Positive"

"392" "Negative"

"393" "Positive"

"394" "Positive"

"395" "Negative"

"396" "Positive"

"397" "Positive"

"398" "Negative"

"399" "Negative"

"400" "Positive"

"401" "Positive"

"402" "Negative"

"403" "Positive"

"404" "Negative"

"405" "Positive"

"406" "Negative"

"407" "Positive"

"408" "Positive"

"409" "Negative"

"410" "Negative"

"411" "Negative"

"412" "Positive"

"413" "Positive"

"414" "Positive"

"415" "Positive"

"416" "Positive"

"417" "Negative"

"418" "Positive"

"419" "Negative"

"420" "Negative"

"421" "Positive"

"422" "Positive"

"423" "Negative"

"424" "Negative"

"425" "Negative"

"426" "Negative"

"427" "Positive"

"428" "Negative"

"429" "Negative"

"430" "Negative"

"431" "Negative"

"432" "Negative"

"433" "Negative"

"434" "Positive"

"435" "Negative"

"436" "Negative"

"437" "Negative"

"438" "Negative"

"439" "Negative"

"440" "Negative"

"441" "Positive"

"442" "Positive"

"443" "Positive"

"444" "Positive"

"445" "Positive"

"446" "Negative"

"447" "Positive"

"448" "Positive"

"449" "Negative"

"450" "Positive"

"451" "Negative"

"452" "Negative"

"453" "Negative"

"454" "Positive"

"455" "Negative"

"456" "Negative"

"457" "Positive"

"458" "Positive"

"459" "Positive"

"460" "Positive"

"461" "Negative"

"462" "Negative"

"463" "Positive"

"464" "Positive"

"465" "Negative"

"466" "Negative"

"467" "Negative"

"468" "Negative"

"469" "Negative"

"470" "Negative"

"471" "Negative"

"472" "Positive"

"473" "Negative"

"474" "Positive"

"475" "Positive"

"476" "Positive"

"477" "Positive"

"478" "Positive"

"479" "Positive"

"480" "Negative"

"481" "Negative"

"482" "Negative"

"483" "Positive"

"484" "Negative"

"485" "Negative"

"486" "Negative"

"487" "Negative"

"488" "Negative"

"489" "Positive"

"490" "Positive"

"491" "Negative"

"492" "Positive"

"493" "Negative"

"494" "Negative"

"495" "Positive"

"496" "Positive"

"497" "Negative"

"498" "Negative"

"499" "Negative"

"500" "Positive"

"501" "Positive"

"502" "Negative"

"503" "Negative"

"504" "Positive"

"505" "Positive"

"506" "Negative"

"507" "Positive"

Answer #1

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Output from minitab is below :

Descriptive Statistics

N | Event | Sample p | 95% CI for p |

507 | 210 | 0.414201 | (0.371324, 0.457078) |

Below is done manually :

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